THE

ANALYTICAL THEORY OF HEAT

JOSEPH FOURIER

, TRANSLATED, WITH NOTES,

BY

ALEXANDER FREEMAN, M.A.,

FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE.

EDITED FOR THE SYNDICS OF THE UNIVERSITY PRESS.

CDambntrge :
AT THE UNIVERSITY PRESS.

LONDON : CAMBRIDGE WAREHOUSE, 17, PATERNOSTER ROW.

CAMBRIDGE: DEIGHTON, BELL, AND CO.
LEIPZIG: F. A. BROCKHAUS.

1878

[All Rights reserved.]

-•'

k

PRINTED BY C. J. CLAY, M.A.,
AT THE UNIVERSITY PRESS.

PREFACE.

IN preparing this version in English of Fourier's
celebrated treatise on Heat, the translator has followed
faithfully the French original. He has, however, ap
pended brief foot-notes, in which will be found references
to other writings of Fourier and modern authors on
the subject : these are distinguished by the initials A. F.
The notes marked R. L. E. are taken from pencil me
moranda on the margin of a copy of the work that
formerly belonged to the late Robert Leslie Ellis,
Fellow of Trinity College, and is now in the possession
of St John's College. It was the translator's hope to
have been able to prefix to this treatise a Memoir
of Fourier's life with some account of his writings ;
unforeseen circumstances have however prevented its
completion in time to appear with the present work.

781452

TABLE

OF

CONTENTS OF THE WORK1.

PAGE

PRELIMINARY DISCOURSE 1

CHAPTER I.

Introduction.

SECTION I.

STATEMENT OF THE OBJECT OF THE WORK.
ART.

I. Object of the theoretical researches .14

2—10. Different examples, ring, cube, sphere, infinite prism ; the variable

temperature at any point whatever is a function of the coordinates
and of the time. The quantity of heat, which during unit of time
crosses a given surface in the interior of the solid, is also a function
of the time elapsed, and of quantities which determine the form and
position of the surface. The object of the theory is to discover these
functions 15

II. The three specific elements which must be observed, are the capacity, the
conducibility proper or permeability, and the external conducibility or
penetrability. The coefficients which express them may be regarded at
first as constant numbers, independent of the temperatures ... 19

12. First statement of the problem of the terrestrial temperatures . . 20
13—15. Conditions necessary to applications of the theory. Object of the

experiments 21

16 — 21. The rays of heat which escape from the same point of a surface

have not the same intensity. The intensity of each ray is proportional

1 Each paragraph of the Table indicates the matter treated of in the articles
indicated at the left of that paragraph. The first of these articles begins at
the page marked on the right.

VI TABLE OF CONTENTS.

ART. PAGE

to the cosine of the angle which its direction makes with the normal to
the surface. Divers remarks, and considerations on the object and extent
of thermological problems, and on the relations of general analysis with
the study of nature 22

SECTION II.

GENERAL NOTIONS AND PRELIMINARY DEFINITIONS.

22 — 24. Permanent temperature, thermometer. The temperature denoted
by 0 is that of melting ice. The temperature of water boiling in a
given vessel under a given pressure is denoted by 1 26

25. The unit which serves to measure quantities of heat, is the heat
required to liquify a certain mass of ice . . . . . . .27

26. Specific capacity for heat ib.

27 — 29. Temperatures measured by increments of volume or by the addi
tional quantities of heat. Those cases only are here considered, in which

the increments of volume are proportional to the increments of the
quantity of heat. This condition does not in general exist in liquids ;
it is sensibly true for solid bodies whose temperatures differ very much
from those which cause the change of state 28

30. Notion of external conducibility ib.

31. We may at first regard the quantity of heat lost as proportional to the
temperature. This proposition is not sensibly true except for certain
limits of temperature . . . . . . . . .29

32 — 35. The heat lost into the medium consists of several parts. The effect

is compound and variable. Luminous heat ib.

36. Measure of the external conducibility . . . . . . . . 31

37. Notion of the conducibility proper. This property also may be observed

in liquids ^

38. 39. Equilibrium of temperatures. The effect is independent of contact . 32
40 — 49. First notions of radiant heat, and of the equilibrium which is

established in spaces void of air ; of the cause of the reflection of rays
of heat, or of their retention in bodies ; of the mode of communication
between the internal molecules; of the law which regulates the inten
sity of the rays emitted. The law is not disturbed by the reflection of

heat . ibt

50, 51. First notion of the effects of reflected heat 37

52 — 56. Remarks on the statical or dynamical properties of heat. It is the
principle of elasticity. The elastic force of aeriform fluids exactly indi
cates their temperatures ....... 39

SECTION III.
PRINCIPLE OF THE COMMUNICATION OF HEAT.

57 — 59. When two molecules of the same solid are extremely near and at
unequal temperatures, the most heated molecule communicates to that
which is less heated a quantity of heat exactly expressed by the product
of the duration of the instant, of the extremely small difference of the
temperatures, and of a certain function of the distance of the molecules . 41

TABLE OF CONTEXTS. Vll

ART. PAGE

60. When a heated body is placed in an aeriform medium at a lower tem
perature, it loses at each instant a quantity of heat which may be
regarded in the first researches as proportional to the excess of the
temperature of the surface over the temperature of the medium . . 43

61 — 64. The propositions enunciated in the two preceding articles are founded
on divers observations. The primary object of the theory is to discover
all the exact consequences of these propositions. We can then measure
the variations of the coefficients, by comparing the results of calculation
with very exact experiments ......... t&.

SECTION IV.
OF THE UNIFORM AND LINEAR MOVEMENT OF HEAT.

65. The permanent temperatures of an infinite solid included between two
parallel planes maintained at fixed temperatures, are expressed by the
equation (v - a) e = (b - a) z ; a and 6 are the temperatures of the two
extreme planes, e their distance, and v the temperature of the section,
whose distance from the lower plane is z . . . ..... 45

66, 67. Notion and measure of the flow of heat ...... 48

68, 69. Measure of the conducibility proper ....... 51

70. Remarks on the case in which the direct action of the heat extends to

a sensible distance ........... 53

71. State of the same solid when the upper plane is exposed to the air . . «6.

72. General conditions of the linear movement of heat ..... 55

SECTION V.

LAW OF THE PERMANENT TEMPERATURES IN A PRISM OF SMALL THICKNESS.

73—80. Equation of the linear movement of heat in the prism. Different

consequences of this equation .... ..... 56

SECTION VI.

THE HEATING OF CLOSED SPACES.

81 — 84. The final state of the solid boundary which encloses the space
heated by a surface 6, maintained at the temperature a, is expressed by
the following equation :

m-n^(a-n)

The value of P is — ( ~ + — + -f- ) , ?n is the temperature of the internal
s \fi K H J

air, n the temperature of the external air, g, h, H measure respectively
the penetrability of the heated surface <r, that of the inner surface of the
boundary s, and that of the external" surface s ; e is the thickness of the
boundary, and K its conducibility proper ....... 62

85, 86. Remarkable consequences of the preceding equation 65

87 — 91. Measure of the quantity of heat requisite to retain at a constant
temperature a body whose surface is protected from the external air by

Vlll TABLE OF CONTENTS.

ABT. PAGE

several successive envelopes. Remarkable effects of the separation of the
surfaces. These results applicable to many different problems . . 67

SECTION VII.
OF THE UNIFOEM MOVEMENT OF HEAT IN THBEE DIMENSIONS.

92, 93. The permanent temperatures of a solid enclosed between six rec
tangular planes are expressed by the equation

v = A + ax + by + cz.

x, y, z are the coordinates of any point, whose temperature is v ; A, a,
b, c are constant numbers. If the extreme planes are maintained by any
causes at fixed temperatures which satisfy the preceding equation, the
final system of all the internal temperatures will be expressed by the

same equation 73

94, 95. Measure of the flow of heat in this prism 75

SECTION VHI.
MEASUKE OF THE MOVEMENT OF HEAT AT A GIVEN POINT OF A GIVEN SOLID.

96 — 99. The variable system of temperatures of a solid is supposed to be
expressed by the equation v—F (x, y, z, t), where v denotes the variable
temperature which would be observed after the time t had elapsed, at the
point whose coordinates are x, y, z. Formation of the analytical expres
sion of the flow of heat in a given direction within the solid ... 78

100. Application of the preceding theorem to the case in which the function

F is e~fft COB x cosy cos z • . . . .82

CHAPTER II.

Equation of the Movement of Heat.

SECTION I.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A RING.

101—105. The variable movement of heat in a ring is expressed by the
equation

dv_K^ d*v hi
di~~CD dy?

The arc x measures the distance of a section from the origin 0 ; v is
the temperature which that section acquires after the lapse of the time t ;
K, C, D, h are the specific coefficients ; S is the area of the section, by
the revolution of which the ring is generated; I is the perimeter of
the section .......... 85

TABLE OF CONTENTS. IX

AET. PAGE

106 — 110. The temperatures at points situated at equal distances are
represented by the terms of a recurring series. Observation of the
temperatures vlt vz, v3 of three consecutive points gives the measure

of the ratio*: We have

The distance between two consecutive points is X, and log w is the decimal
logarithm of one of the two values of w . . . . . . .86

SECTION II.
EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID SPHERE.

Ill — 113. x denoting the radius of any shell, the movement of heat in the
sphere is expressed by the equation

dv K d*v 2dv

114— 117. Conditions relative to the state of the surface and to the initial

state of the solid 92

SECTION IH.
EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID CYLINDER. ^X

118 — 120. The temperatures of the solid are determined by three equations;
the first relates to the internal temperatures, the second expresses the
continuous state of the surface, the third expresses the initial state of
the solid 95

SECTION IV.

EQUATIONS OF THE VARIED MOVEMENT OF HEAT IN A SOLID PRISM
OF INFINITE LENGTH.

121 — 123. The system of fixed temperatures satisfies the equation

d^v d^v d2v
dtf+dfi + d^= ;

v is the temperature at a point whose coordinates are x, y, z . . . 97
124, 125. Equation relative to the state of the surface and to that of the

first section 99

SECTION V.

EQUATIONS OF THE. VARIED MOVEMENT OF HEAT IN A SOLID CUBE.

126—131. The system of variable temperatures is determined by three
equations ; one expresses the internal state, the second relates to the

t state of the surface, and the third expresses the initial state . . . 101
"

TABLE OF CONTENTS.

SECTION VI.

GENERAL EQUATION OF THE PROPAGATION OF HEAT IN THE INTERIOR
OF SOLIDS.

ART. PAGE

132—139. Elementary proof of properties of the uniform movement of heat
in a solid enclosed between six orthogonal planes, the constant tem
peratures being expressed by the linear equation,

v = A - ax - by - cz.

The temperatures cannot change, since each point of the solid receives
as much heat as it gives off. The quantity of heat which during the
unit of time crosses a plane at right angles to the axis of z is the same,
through whatever point of that axis the plane passes. The value of this
common flow is that which would exist, if the coefficients a and 6

were nul 104

140, 141. Analytical expression of the flow in the interior of any solid. The

equation of the temperatures being v=f(x, y, z, t) the function -Ku —

expresses the quantity of heat which during the instant dt crosses an
infinitely small area w perpendicular to the axis of z, at the point whose
coordinates are x, ?/, z, and whose temperature is v after the time t

has elapsed 109

142 — 145. It is easy to derive from the foregoing theorem the general
equation of the movement of heat, namely

dv K

SECTION VII.

GENERAL EQUATION BELATIVE TO THE SURFACE.

146 — 154. It is proved that the variable temperatures at points on the
surface of a body, which is cooling in air, satisfy the equation

dv dv dv h

being the differential equation of the surface which bounds the solid,
and q being equal to (m? + n*+p'*)2. To discover this equation we
consider a molecule of the envelop which bounds the solid, and we express
the fact that the temperature of this element does not change by a finite
magnitude during an infinitely small instant. This condition holds and
continues to exist after that the regular action of the medium has been
exerted during a very small instant. Any form may be given to the
element of the envelop. The case in which the molecule is formed by
rectangular sections presents remarkable properties. In the most simple
case, which is that in which the base is parallel to the tangent plane,
the truth of the equation is evident ..... 115

TABLE OF CONTENTS. XI

SECTION VIII.

APPLICATION OF THE GENERAL EQUATIONS.

ART. PAGE

155, 156. In applying the general equation (A) to the case of the cylinder
and of the sphere, we find the same equations as those of Section III.
and of Section II. of this chapter 123

SECTION IX.

GENERAL BEMARKS.

157—162. Fundamental considerations on the nature of the quantities
x, t, r, K, h, C, D, which enter into all the analytical expressions of the
Theory of Heat. Each of these quantities has an exponent of dimension
which relates to the length, or to the duration, or to the temperature.
These exponents are found by making the units of measure vary . . 126

CHAPTER III.

Propagation of Heat in an infinite rectangular solid.

SECTION I.

STATEMENT OF THE PROBLEM.

163—166. The constant temperatures of a rectangular plate included be
tween two parallel infinite sides, maintained at the temperature 0, are

expressed by the equation -^ + -^=0 131

167 — 170. If we consider the state of the plate at a very great distance from
the transverse edge, the ratio of the temperatures of two points whose
coordinates are a^, y and xz,y changes according as the value of y
increases ; xl and x.2 preserving their respective values. The ratio has
a limit to which it approaches more and more, and when y is infinite,
it is expressed by the product of a function of x and of a function of y.
This remark suffices to disclose the general form of v, namely,

^ = S):iV~(2<~1)a:. cos(2i-l).y.

It is easy to ascertain how the movement of heat in the plate is
effected 134

Xll TABLE OF CONTENTS.

SECTION II.

FIBST EXAMPLE OF THE USE OF TRIGONOMETRIC SERIES IN THE
THEORY OF HEAT.

ART. PAGE

171 — 178. Investigation of the coefficients in the equation

l=a cos x +* cos 3x + ecos 5x + d cos 7x + etc.
From which we conclude

or -r=coso:-5cos3a!: + £eos5a5- = cos7#-t- etc.

£ o O i

SECTION III.
REMARKS ON THESE SERIES.

179—181. To find the value of the series which forms the second member,
the number m of terms is supposed to be limited, and the series becomes
a function of x and m. This function is developed according to powers of
the reciprocal of m, and m is made infinite ......

182—184. The same process is applied to several other series . . .

185 — 188. In the preceding development, which gives the value of the
function of x and m, we determine rigorously the limits within which the
sum of all the terms is included, starting from a given term , . .

189. Very simple process for forming the series

SECTION IV.

GENERAL SOLUTION.

190, 191. Analytical expression of the movement of heat in a rectangular
slab ; it is decomposed into simple movements .....

192 — 195. Measure of the quantity of heat which crosses an edge or side
parallel or perpendicular to the base. This expression of the flow suffices
to verify the solution

196—199. Consequences of this solution. The rectangular slab must be
considered as forming part of an infinite plane ; the solution expresses
the permanent temperatures at all points of this plane . . . .

200—204. It is proved that the problem proposed admits of no other solu
tion different from that which we have just stated ....

TABLE OF CONTENTS. Xlll

SECTION V.
FINITE EXPRESSION OF THE RESULT OP THE SOLUTION.

ART. PAGE

205, 206. The temperature at a point of the rectangular slab \vhose co
ordinates are x and y, is expressed thus

SECTION VI.
DEVELOPMENT OF AN ARBITRARY FUNCTION IN TRIGONOMETRIC SERIES.

207 — 214. The development obtained by determining the values of the un
known coefficients in the following equations infinite in number :

A =

C = a + 25b + 35c + ±5d + &c.f
D = a + 2'b + 37c + 47d + Ac.,
Ac., &c.

To solve these equations, we first suppose the number of equations to be
m, and that the number of unknowns a, b, c, d, &c. is m only, omitting
all the subsequent terms. The unknowns are determined for a certain
value of the number ni, and the limits to which the values of the coeffi
cients continually approach are sought; these limits are the quantities
which it is. required to determine. Expression of the values of a, 6, c, d,
&G. when m is infinite ......... • 168

215, 216. The function $(x) developed under the form

sin2o; + c

which is first supposed to contain only odd powers of x . . . .179
217, 218. Different expression of the same development. Application to the

function ex - e~x . . . ..... . . . 181

219 — 221. Any function whatever <p(x) may be developed under the form

^ sin£ + a2 sin^x + Og sin3.z+ ... +0^ sin j'x + Ac.
The value of the general coefficient a< is - / dx <f> (x) sin ix. Whence we

7T J 0

derive the very simple theorem
^ <£(«) = sin a: /""da 0{a) sina -f sm2xj ^da^a) sin2a + sin3a; /""da^a) sin3a + &c.,

IT f=3° . r1*

whence — 0(x) = S sin ix / da<f>(a.) sin fa .... 184

2 t=i J o

222, 223. Application of the theorem : from it is derived the remarkable
series,

- cos x = — sin x + — sin 4.r + — sin 7x + -— sin 9^; + &c. . . 188

*i . A *9« • ' . • ' D.I •• v

xiv TABLE OF CONTENTS.

ART. PAGE

224, 225. Second theorem on the development of functions in trigono
metrical series :

-^(o5)=S cosix rndacosia\!/(a).

* i=0 Jo

Applications : from it we derive the remarkable series
1 . t 1 cos2x cos 4x

226 — 230. The preceding theorems are applicable to discontinuous functions,
and solve the problems which are based upon the analysis of Daniel
Bernoulli in the problem of vibrating cords. The value of the series,

sin x versin a + ~ ski 2x versin 2 a + ^ sin 3x versin 3 a -f &c. ,

is ^ , if we attribute to # a quantity greater than 0 and less than a; and

the value of the series is 0, if x is any quantity included between a and |TT.
Application to other remarkable examples ; curved lines or surfaces which
coincide in a part of their course, and differ in all the other parts . . 193

231 — 233. Any function whatever, F(x), may be developed in the form

.
p) + ^ sina; + Z>2 sin 2» -f 63 sin 3a + &c.

Each of the coefficients is a definite integral. We have in general
2irA = f*"dx F(x) , ira< = f*JdxF(x) cos ix,

and irbt — f dx F(x) sin ix.

We thus form the general theorem, which is one of the chief elements of
our analysis :

i=^+co / .,J.jj» /*.X«f X

2irF(x) = S (cos ix I daF(a) cos ia + sin ix J daF(a) sin ia ) ,

i=— eo \ J —TT •— If J

i=+oo P + ir

or 2irF(x) = 2 I daF(a)coa(ix-id) 199

»=_„•'-«•

234. The values of F(x) which correspond to values of x included
between - TT and + TT must be regarded as entirely arbitrary. We may
also choose any limits whatever for ic ....... 204

235. Divers remarks on the use of developments in trigonometric series . 206

SECTION VII.
APPLICATION TO THE ACTUAL PEOBLEM.

236. 237. Expression of the permanent temperature in the infinite rectangular
slab, the state of the transverse edge being represented by an arbitrary
function .... 209

TABLE OF CONTENTS. XV

CHAPTER IV.

Of the linear and varied Movement of Heat in a ring.

SECTION I.

GENERAL SOLUTION OF THE PROBLEM.

ART. PAGE

238—241. The variable movement which we are considering is composed of
simple movements. In each of these movements, the temperatures pre
serve their primitive ratios, and decrease with the time, as the ordinates v
of a line whose equation is v=A. e~mt. Formation of the general ex
pression ... 213

242 — 244. Application to some remarkable examples. Different consequences

of the solution 218

245, 246. The system of temperatures converges rapidly towards a regular
and final state, expressed by the first part of the integral. The sum of
the temperatures of two points diametrically opposed is then the same,
whatever be the position of the diameter. It is equal to the mean tem
perature. In each simple movement, the circumference is divided by
equidistant nodes. All these partial movements successively disappear,
except the first ; and in general the heat distributed throughout the solid
assumes a regular disposition, independent of the initial state . . 221

SECTION II.
OP THE COMMUNICATION OF HEAT BETWEEN SEPARATE MASSES.

247 — 250. Of the communication of heat between two masses. Expression
of the variable temperatures. Remark on the value of the coefficient

which measures the conducibility 225

251 — 255. Of the communication of heat between n separate masses, ar
ranged in a straight line. Expression of the variable temperature of each
mass; it is given as a function of the time elapsed, of the coefficient
which measures the couducibility, and of all the initial temperatures

regarded as arbitrary 228

256, 257. Remarkable consequences of this solution 236

258. Application to the case in which the number of masses is infinite . . 237
259 — 266. Of the communication of heat between n separate masses arranged
circularly. Differential equations suitable to the problem ; integration of
these equations. The variable temperature of each of the masses is ex
pressed as a function of the coefficient which measures the couducibility,
of the time which has elapsed since the instant when the communication
began, and of all the initial temperatures, which are arbitrary ; but in
order to determine these functions completely, it is necessary to effect

the elimination of the coefficients 238

267—271. Elimination of the coefficients in the equations which contain

these unknown quantities and the given initial temperatures . . . 247

XVI TABLE OF CONTENTS.

ART. PAGE

272, 273. Formation of the general solution : analytical expression of the

result 253

274 — 276. Application and consequences of this solution .... 255
277, 278. Examination of the case in which the number n is supposed infinite.
We obtain the solution relative to a solid ring, set forth in Article 241,
and the theorem of Article 234. We thus ascertain the origin of the
analysis which we have employed to solve the equation relating to con
tinuous bodies 259

279. Analytical expression of the two preceding results .... 262
280 — 282. It is proved that the problem of the movement of heat in a ring

admits no other solution. The integral of the equation -^= k -=-? is

dt dx*

evidently the most general which can be formed « « « « . 263

CHAPTER V.

Of the Propagation of Heat in a solid sphere.

SECTION I.
GENEBAL SOLUTION.

283 — 289. The ratio of the variable temperatures of two points in the solid
is in the first place considered to approach continually a definite limit.

This remark leads to the equation v=A g«-J&i%| which expresses

the simple movement of heat in the sphere. The number n has an

infinity of values given by the definite equation — - — - = 1 - hX. The

tan nX

radius of the sphere is denoted by X, and the radius of any concentric
sphere, whose temperature is v after the lapse of the time t, by x\ h
and K are the specific coefficients; A is any constant. Constructions
adapted to disclose the nature of the definite equation, the limits and

values of its roots 268

290 — 292. Formation of the general solution ; final state of the solid . . 274
293. Application to the case in which the sphere has been heated by a pro
longed immersion ,..,.. 277

SECTION n.

DlFFEBENT BEMABKS ON THIS SOLUTION.

294 — 296. Kesults relative to spheres of small radius, and to the final tem
peratures of any sphere ...... 279

298—300. Variable temperature of a thermometer plunged into a liquid
which is cooling freely. Application of the results to the comparison and
use of thermometers , , 282

TABLE OF CONTENTS. XV11

ART. PAGB

301. Expression of the mean temperature of the sphere as a function of the

time elapsed 286

302 — 304. Application to spheres of very great radius, and to those in which

the radius is very small 287

305. Kernark on the nature of the definite equation which gives all the values

of n . ,289

CHAPTER VI.

Of the Movement of Heat in a solid cylinder.

306, 307. We remark in the first place that the ratio of the variable tem
peratures of two points of the solid approaches continually a definite
limit, and by this we ascertain the expression of the simple movement.
The function of x which is one of the factors of this expression is given
by a differential equation of the second order. A number g enters into
this function, and must satisfy a definite equation 291

308, 309. Analysis of this equation. By means of the principal theorems of

algebra, it is proved that all the roots of the equation are real . . . 294

310. The function u of the variable x is expressed by

i r1* i—

u =— / dr cos (xtjg sin r) ;

and the definite equation is hu + — =0, giving to x its complete value X. 296
311, 312. The development of the function $(z) being represented by

22 , 2

"f&C>'

the value of the series

c<2 et*

22 22. 42 22. 42. 62

1 t*

is — / dii(f>(tsmu).

irJ Q

Remark on this use of definite integrals ....... 298

313. Expression of the function u of the variable a; as a continued fraction . 300

314. Formation of the general solution 301

315 — 318. Statement of the analysis which determines the values of the co
efficients 303

319. General solution 308

320. Consequences of the solution . . 309

XVI 11 TABLE OF CONTENTS.

CHAPTER VII.

Propagation of Heat in a rectangular prism.

ART. PAGE

321 — 323. Expression of the simple movement determined by the general
properties o£ he^t, ar^d by the form of the solid. Into this expression
enters an arc e which satisfies a transcendental equation, all of whose
roots are real 311

324. All the unknown coefficients are determined by definite integrals . 313

325. General solution of the problem ........ 314

326. 327. The problem proposed admits no other solution .... 315
328, 329. Temperatures at points on the axis of the prism .... 317

330. Application to the case in which the thickness of the prism is very
small 318

331. The solution shews how the uniform movement of heat is established

in the interior of the solid 319

332. Application to prisms, the dimensions of whose bases are large . . 322

CHAPTER VIII.

Of the Movement of Heat in a solid cube.

333, 334. Expression of the simple movement. Into it enters an arc e

which must satisfy a trigonometric equation all of whose roots are real . 323

335, 336. Formation of the general solution . 324

337. The problem can admit no other solution . . . . . . 327

338. Consequence of the solution ib.

339. Expression of the mean temperature 328

340. Comparison of the final movement of heat in the cube, with the
movement which takes place in the sphere 329

341. Application to the simple case considered in Art. 100 .... 331

CHAPTER IX.

Of the Diffusion of Heat.
SECTION I.

OF THE FHEE MOVEMENT OF HEAT IN AN INFINITE LlNE.

342 — 347. We consider the linear movement of heat in an infinite line, a
part of which has been heated; the initial state is represented by
v — F(x). The following theorem is proved :

fl °°
dq cos qx I da F(a) cos ga.
'o

TABLE OF CONTENTS. XIX

ABT. PAGE

The function P (x) satisfies the condition F (x) = F ( - x). Expression of

the variable temperatures .......... 333

348. Application to the <case in which all the points of the part heated
have received the same initial temperature. The integral

I — sin 2 cos qx is i
Jo

—

if we give to x a value included between 1 and - 1.

The definite integral has a nul value, if a; is not included between
1 and - 1 ............. 338

3-49. Application to the case in which the heating given results from the

final state which the action of a source of heat determines . . . 339
350. Discontinuous values of the function expressed by the integral

34°

351 — 353. We consider the linear movement of heat in a line whose initial
temperatures are represented by v—f(x) at the distance x to the right
of the origin, and by v = -f(x) at the distance x to the left of the origin.
Expression of the variable temperature at any point. The solution
derived from the analysis which expresses the movement of heat in an
infinite line ..... ...... . ib.

354. Expression of the variable temperatures when the initial state of the

part heated is expressed by an entirely arbitrary function . . . 343

355 — 358. The developments of functions in sines or cosines of multiple arcs

are transformed into definite integrals ....... 345

359. The following theorem is proved :

!Lf(x}— I dqsinqx I daf (a) sinqa.
* Jo Jo

The function / (x) satisfies the condition :

348

360 — 362. Use of the preceding results. Proof of the theorem expressed
by the general equation :

f+» r*

7T0 (x) = I da <p (a) I dq COS (qx - qa).
./-» Jo

This equation is evidently included in equation (II) stated in Art. 234.
(See Art. 397) ib.

363. The foregoing solution shews also the variable movement of heat in an
infinite line, one point of which is submitted to a constant temperature . 352

364. The game problem may also be solved by means of another form of the
integral. Formation of this integral 354

365. 366. Application of the solution to an infinite prism, whose initial
temperatures are nul. Remarkable consequences 356

367 — 369. The same integral applies to the problem of the diffusion of heat.
The solution which we derive from it agrees with that which has been
stated in Articles 347, 348 .... .... 362

XX TABLE OF CONTENTS.

ART.

370, 371. Bemarks on different forms of the integral of the equation

du d?u

SECTION II.

OF THE FEEE MOVEMENT OF HEAT IN AN INFINITE SOLID.

372 — 376. The expression for the variable movement of heat in an infinite
solid mass, according to three dimensions, is derived immediately from
that of the linear movement. The integral of the equation

dv _ d?v d2v d2v
Tt ~ dx* + dy* + <P

solves the proposed problem. It cannot have a more extended integral j
it is derived also from the particular value

v = e~n2t cos nx,
or from this :

which both satisfy the equation — = — ^ . The generality of the in-

tegrals obtained is founded upon the following proposition, which may be
regarded as self-evident. Two functions of the variables x, y, z, t are
necessarily identical, if they satisfy the differential equation

dv dsv dzv dsv
dt = dx? + dy* + ~dz? '

and if at the same time they have the same value for a certain value
of t ....... " .......

377 — 382. The heat contained in a part of an infinite prism, all the other
points of which have nul initial temperature, begins to be distributed
throughout the whole mass ; and after a certain interval of time, the
state of any part of the solid depends not upon the distribution of the
initial heat, but simply upon its quantity. The last result is not due
to the increase of the distance included between any point of the mass
and the part which has been heated; it is entirely due to the increase
of the time elapsed. In all problems submitted to analysis, the expo
nents are absolute numbers, and not quantities. We ought not to omit
the parts of these exponents which are incomparably smaller than the
others, but only those whose absolute values are extremely small .

383 — 385. The same remarks apply to the distribution of heat in an infinite
solid . * * t ...... ....

SECTION HI.

THE HIGHEST TEMPERATURES IN AN INFINITE SOLID.

386, 387. The heat contained in part of the prism distributes itself through
out the whole mass. The temperature at a distant point rises pro
gressively, arrives at its greatest value, and then decreases. The time

TABLE OF CONTENTS.

ART. PAGE

after which this maximum occurs, is a function of the distance x.
Expression of this function for a prism whose heated points have re
ceived the same initial temperature 385

388—391. Solution of a problem analogous to the foregoing. Different

results of the solution 387

392 — 395. The movement of heat in an infinite solid is considered ; and
the highest temperatures, at parts very distant from the part originally
heated, are determined 392

SECTION IV.
COMPARISON OF THE INTEGRALS.

396. First integral (a) of the equation -=- = -=-£ (a). This integral expresses

the movement of heat in a ring ...... . . 396

397. Second integral (/3) of the same equation (a). It expresses the linear
movement of heat in an infinite solid ....... 398

398. Two other forms (7) and (5) of the integral, which are derived, like the
preceding form, from the integral (a) ....... t'6.

399. 400. First development of the value of v according to increasing powers
of the time t. Second development according to the powers of v. The
first must contain a single arbitrary function of t ..... 399

401. Notation appropriate to the representation of these developments. The
analysis which is derived from it dispenses with effecting the develop
ment in series ............ 402

402. Application to the equations :

d-v d*v d2v . dzv d*v ,

^ = d*+d?--:"-(e)l •ndd?+^=° ...... (d)- • • 404

403. Application to the equations :

(/) • • • • 405

404. Use of the theorem E of Article 361, to form the integral of equation (/)

of the preceding Article .......... 407

405. Use of the same theorem to form the integral of equation (d) which
belongs to elastic plates ......... k 409

406. Second form of the same integral ........ 412

407. Lemmas which serve to effect these transformations .... 413

408. The theorem expressed by equation (E), Art. 361, applies to any number

of variables ......... ... '415

409. Use of this proposition to form the integral of equation (c) of Art. 402 . 416

410. Application of the same theorem to the equation

d2v d-v d-v

+ + = ° ...... 41S

xxii TABLE OF CONTENTS.

ART.

411. Integral of equation (e) of vibrating elastic surfaces .... 419

412. Second form of the integral 421

413. Use of the same theorem to obtain the integrals, by summing the
series which represent them. Application to the equation

dv dzv

Integral under finite form containing two arbitrary functions of t . . 422

414. The expressions change form when we use other limits of the definite
integrals 425

415. 416. Construction which serves to prove the general equation

417. Any limits a and b may be taken for the integral with respect to a.
These limits are those of the values of x which correspond to existing
values of the function f(x). Every other value of x gives a nul result
forf(x) 429

418. The same remark applies to the general equation

the second member of which represents a periodic function . . . 432

419. The chief character of the theorem expressed by equation (#) consists
in this, that the sign / of the function is transferred to another unknown

a, and that the chief variable x is only under the symbol cosine . . 433

420. Use of these theorems in the analysis of imaginary quantities . . 435

421. Application to the equation -^ + ^4 = 0 . . . . . .436

dx* dy*

422. General expression of the fluxion of the order t,

423. Construction which serves to prove the general equation. Consequences
relative to the extent of equations of this kind, to the values of / (x)
which correspond to the limits of x, to the infinite values of f(x). . 438

424 — 427. The method which consists in determining by definite integrals
the unknown coefficients of the development of a function of x under
the form

is derived from the elements of algebraic analysis. Example relative to
the distribution of heat in a solid sphere. By examining from this
point of view the process which serves to determine the coefficients, we
solve easily problems which may arise on the employment of all the terms
of the second member, on the discontinuity of functions, on singular or
infinite values. The equations which are obtained by this method ex
press either the variable state, or the initial state of masses of infinite
dimensions. The form of the integrals which belong to the theory of

TABLE OF CONTENTS. xxiii

ART. PAGB

heat, represents at the same time the composition of simple movements,
and that of an infinity of partial effects, due to the action of all points of
the solid 441

428. General remarks on the method which has served to solve the analytical
problems of the theory of heat 450

429. General remarks on the principles from which we have derived the dif
ferential equations of the movement of heat 456

430. Terminology relative to the general properties of heat .... 462

431. Notations proposed 463

432. 433. General remarks on the nature of the coefficients which enter into

the differential equations of the movement of heat . . . . . 464

ERRATA.

Page 9, line 28, for III. read IV.

Pages 54, 55, for k read K.

Page 189, line 2, The equation should be denoted (A).

Page 205, last line but one, for x read A'.

Page 298, line 18, for ~ read ^.
dr dx

Page 299, line 16, for of read in.
,, ,, last line, read

r

du 0 (t sin w) =

Page 300, line 3, for Az, 44, A6, read irA^, 7ivt4, ^^A^
Page 407, line 12, for d<f> read dp.
Page 432, line 13, read (x-a).

CORRECTIONS to the Edition of Fourier's Analytical Theory of
Heat, by A. FREEMAN, M.A., Cambridge, 1878.

PAGE LINE ERROR

9 28 III.

19 10 et pa*sim. Conductivity

— 14 ratio of their capacities

26

solids and liquids increase in
volume

27 27

dissolve

28 2

occupy

°,. \tlirough § 71, A-
oo 1

57 4

right

58

§§ 73—80, A-

66 6

as

11

m — n = a

67 31

to

72 4

j(l + Jf]

13

in denominator

14

j

90 16

V2

146 9

cos (2m - 3x)

20

1

22 m3

152 17

+

26

±K

K

27

x

29

156 11

e~5x cos oy

162 21

<f>(x, y)

164 22

1

CORRECTION

IV.

Conductivity

inverse ratio of their capacities

add, in most cases

melt
occupy

K

add, or left (pro re nata)

K

omit, as

add, 1

cos (2m -3)x
1

+ K
K

K

if K is defined

- e-5a; cos oy

CORRECTIONS.

PAGE LINE ERROR

169 14 B

14 D

172 2 — 4 the numerals

10 62

26 A.2

174 30 1222324252

180 last remove (A)

181 23 216

1
2

182 9
184 18
189 2

— — denote the equation by

194
195

18
12

1

2*

2

- when -/is even

CORRECTION

-B
-D

should be squared

52

1*2, 3, 4, 5

to end of line 11
215.
1

s

1
I*

(A), for sake of note p. 191

2
_ i)»_ when i is of form 2n+ 1

220 31 27r
— 36 2-n-rM

221 through § 245, «0

226 17 a + (a-/3)

cos x cos 3x cos 5x - cos 7^

14

within the uracKets

(7T\2

A2 83

—

note

2 ) ~X

(2) ~x^

205

17

X

X

210

5

2r

r

211

15

in value of 2F (y, p)

insert + before e-(s>-v^-V

212

15

proportianal

proportional

216

6, 7

F

f t

_

16

e ^

e r"

218

28

heat

temperature

29

within the brackets

the signs of all terms +

&

—

—

instead of M

its value, — (1 - e~2w)

7T

.

"*"

ZwrMCDS

227 23 - 3 , twice

&

228 13—17 k

229 18 /3+(a-/3)^
232 5 B

6 tiNsinmu, and ?i-sm(m-
smw v

_^_ sin mu, and

sin (m - 1) u

'

CORRECTIONS.

PAGE LINE

232
239
270
271

284

16 5 + 2 = cost*

2

20
3
4

a
oM

I - hX
0-006500

286 14

— last 2niX-&e.

295 l dmV

300

3

^9,

304

3

«2«1

—

18

^

—

22

/::

307

18

/**

309

6

w

311

13

Jv

du

dx

(v) dx

x" ~T~
dz

— 16 the order of the equations

313 27 n tan vl

324 29, 30 comma after bracket

325 25, 26 do.

CORRECTION
q + 2 = 2 COS U
a

olul
1-hX
0-006502

71.Z-&C.

da;2

I

vx) dx

dv
dz

should be interchanged
n tan nl
dele comma
do.

326

8

r*=£

,,dX

dt

dt

—

13

= , thrice
A

h

thrice

335

27

after

at

336

6

k — -

a

%

dz*

a

!*2

—

8

qlt q2j q^

'

—

9

flj, a.2, a3

O«. fln

337

20

cosqx

cos qjX

339

26

Q

l<

?

—

27

Q

r<

>

341

20

Ul

w

23

a sin —

/7 e

. 1TX

X

U-jC

LnT

345

2

cosqa

sin

fM

355

15

n

Wi

4 CORRECTIONS.

PAGE LINE ERROR CORRECTION

355 16 a bracket is missing

356 5 sign of last term should be +

— 12 sign of first term should be +

— 25 c-9 e~*

_HLt _HLt_

359 5 e GDS ue CDS

360 23 0-00 i'00
362 18 e-u e~ht

372 1 >/TT *w f/ie denominator should be in the numerator

392 2 I S

396 3 3 in numerator 3$

407 12 d(f> dP

— 28 equation integration
432 13 (a; -a) (^-a)

The Editor takes this opportunity of expressing his thanks
to ROBERT E. BAYNES, Esq. and to WALTER G. WOOLCOMBE, Esq.
for the majority of these corrections.

ADDENDUM. An article "Ow the linear motion of heat, Part II.", written by
Sir WM THOMSON under the signature N.N., will be found in the Cambridge
Mathematical Journal, Vol. III. pp. 206—211, and in Vol. I. of the Author's
collected writings. It examines the conditions, subject to which an arbitrary dis
tribution of heat in an infinite solid, bounded by a plane, may be supposed to
have resulted, by conduction, in course of time, from some previous distribu
tion. [A. F.]

MURSTON RECTORY, SITTINGBOURNE, KENT.
June 21st, 1888.

PEELIMINARY DISCOURSE.

PRIMARY causes are unknown to us; but are subject to simple
and constant laws, which may be discovered by observation, the
study of them being the object of natural philosophy.

Heat, like gravity, penetrates every substance of the universe,
its rays occupy all parts of space. The object of our work is to
set forth the mathematical laws which this element obeys. The
theory of heat will hereafter form one of the most important
branches of general physics.

The knowledge of rational mechanics, which the most ancient
nations had been able to acquire, has not come down to us, and
the history of this science, if we except the first theorems in
harmony, is not traced up beyond the discoveries of Archimedes.
This great geometer explained the mathematical principles of
the equilibrium of solids and fluids. About eighteen centuries
elapsed before Galileo, the originator of dynamical theories, dis
covered the laws of motion of heavy bodies. Within this new
science Newton comprised the whole system of the universe. The
successors of these philosophers have extended these theories, and
given them an admirable perfection: they have taught us that
the most diverse phenomena are subject to a small number of
fundamental laws which are reproduced in all the acts of nature.
It is recognised that the same principles regulate all the move
ments of the stars, their form, the inequalities of their courses,
the equilibrium and the oscillations of the seas, the harmonic
vibrations of air and sonorous bodies, the transmission of light,
capillary actions, the undulations of fluids, in fine the most com
plex effects of all the natural forces, and thus has the thought

F. H. 1

2 THEORY OF HEAT.

0

of Newton been confirmed : quod tarn paucis tarn multa prcestet
geometria gloriatur\

But whatever may be the range of mechanical theories, they
do not apply to the effects of heat. These make up a special
order of phenomena, which cannot be explained by the principles
of motion and equilibrium. We have for a long time been in
possession of ingenious instruments adapted to measure many
of these effects; valuable observations have been collected ; but
in this manner partial results only have become known, and
not the mathematical demonstration of the laws which include
them all.

I have deduced these laws from prolonged study and at
tentive comparison of the facts known up to this time : all these
facts I have observed afresh in the course of several years with
the most exact instruments that have hitherto been used.

To found the theory, it was in the first place necessary to
distinguish and define with precision the elementary properties
which determine the action of heat. I then perceived that all the
phenomena which depend on this action resolve themselves into
a very small number of general and simple facts ; whereby every
physical problem of this kind is brought back to an investiga
tion of mathematical analysis. From these general facts I have
concluded that to determine numerically the most varied move
ments of heat, it is sufficient to submit each substance to three
fundamental observations. Different bodies in fact do not possess
in the same degree the power to contain heat, to receive or transmit
it across their surfaces, nor to conduct it through the interior of
their masses. These are the three specific qualities which our
theory clearly distinguishes and shews how to measure.

It is easy to judge how much these researches concern the
physical sciences and civil economy, and what may be their
influence on the progress of the arts which require the employ
ment and distribution of heat. They have also a necessary con
nection with the -system of the world, and their relations become
known when we consider the grand phenomena which take place
near the surface of the terrestrial globe.

1 Phiiosophia naturalis principia mathematica. Auctoris prafatio ad lectorem.
Ac gloriatur geoinetria quod tarn paucis principiis aliunde petitis tarn multa
proestet. [A. F.]

PRELIMINARY DISCOURSE. 3

In fact, the radiation of the sun in which this planet is
incessantly plunged, penetrates the air, the earth, and the waters ;
its elements are divided, change in direction every way, and,
penetrating the mass of the globe, would raise its mean tem
perature more and more, if the heat acquired were not exactly
balanced by that which escapes in rays from all points of the
surface and expands through the sky.

Different climates, unequally exposed to the action of solar
heat, have, after an immense time, acquired the temperatures
proper to their situation. This effect is modified by several ac
cessory causes, such as elevation, the form of the ground, the
neighbourhood and extent of continents and seas, the state of the
surface, the direction of the winds.

The succession of day and night, the alternations of the
seasons occasion in the solid earth periodic variations, which are
repeated every day or every year: but these changes become
less and less sensible as the point at which they are measured
recedes from the surface. No diurnal variation can be detected
at the depth, of about three metres [ten feet] ; and the annual
variations cease to be appreciable at a depth much less than
sixty metres. The temperature at great depths is then sensibly
fixed at a given place : but it is not the same at all points of the
same meridian ; in general it rises as the equator is approached.

The heat which the sun has communicated to the terrestrial
globe, and which has produced the diversity of climates, is now
subject to a movement which has become uniform. It advances
within the interior of the mass which it penetrates throughout,
and at the same time recedes from the plane of the equator, and
proceeds to lose itself across the polar regions.

In the higher regions of the atmosphere the air is very rare
and transparent, and retains but a minute part of the heat of
the solar rays : this is the cause of the excessive cold of elevated
places. The lower layers, denser and more heated by the land
and water, expand and rise up : they are cooled by the very
fact of expansion. The great movements of the air, such as
the trade winds which blow between the tropics, are not de
termined by the attractive forces of the moon and sun. The
action of these celestial bodies produces scarcely perceptible
oscillations in a fluid so rare and at so great a distance. It

1—2

4 THEORY OF HEAT.

is the changes of temperature which periodically displace every
part of the atmosphere.

The waters of the ocean are differently exposed at their
surface to the rays of the sun, and the bottom of the basin
which contains them is heated very unequally from the poles
to the equator. These two causes, ever present, and combined
with gravity and the centrifugal force, keep up vast movements
in the interior of the seas. They displace and mingle all the
parts, and produce those general and regular currents which
navigators have noticed.

Radiant heat which escapes from the surface of all bodies,
and traverses elastic media, or spaces void of air, has special
laws, and occurs with widely varied phenomena. The physical
explanation of many of these facts is already known ; the mathe
matical theory which I have formed gives an exact measure of
them. It consists, in a manner, in a new catoptrics which
has its own theorems, and serves to determine by analysis all
the effects of heat direct or reflected.

The enumeration of the chief objects of the theory sufficiently
shews the nature of the questions which I have proposed to
myself. What are the elementary properties which it is requisite
to observe in each substance, and what are the experiments
most suitable to determine them exactly? If the distribution
of heat in solid matter is regulated by constant laws, what is
the mathematical expression of those laws, and by what analysis
may we derive from this expression the complete solution of
the principal problems ? Why do terrestrial temperatures cease
to be variable at a depth so small with respect to the radius
of the earth ? Every inequality in the movement of this planet
necessarily occasioning an oscillation of the solar heat beneath
the surface, what relation is there between the duration of its
period, and the depth at which the temperatures become con
stant ?

What time must have elapsed before the climates could acquire
the different temperatures which they now maintain; and what
are the different causes which can now vary their mean heat ?
Why do not the annual changes alone in the distance of the
sun from the earth, produce at the surface of the earth very
considerable changes in the temperatures ?

PRELIMINARY DISCOURSE. 5

From what characteristic can we ascertain that the earth
has not entirely lost its original heat; and what are the exact
laws of the loss ?

If, as several observations indicate, this fundamental heat
is not wholly dissipated, it must be immense at great depths,
and nevertheless it has no sensible influence at the present time
on the mean temperature of the climates. The effects which
are observed in them are due to the action of the solar rays.
But independently of these two sources of heat, the one funda
mental and primitive, proper to the terrestrial globe, the other due
to the presence of the sun, is there not a more universal cause,
which determines the temperature of the heavens, in that part
of space which the solar system now occupies? Since the ob
served facts necessitate this cause, what are the consequences
of an exact theory in this entirely new question; how shall we
be able to determine that constant value of the temperature of
space, and deduce from it the temperature which belongs to each
planet ?

To these, questions must be added others which depend on
the properties of radiant heat. The physical cause of the re
flection of cold, that is to say the reflection of a lesser degree
of heat, is very distinctly known ; but what is the mathematical
expression of this effect ?

On what general principles do the atmospheric temperatures
depend, whether the thermometer which measures them receives
the solar rays directly, on a surface metallic or unpolished,
or whether this instrument remains exposed, during the night,
under a sky free from clouds, to contact with the air, to radiation
from terrestrial bodies, and to that from the most distant and
coldest parts of the atmosphere ?

The intensity of the rays which escape from a point on the
surface of any heated body varying with their inclination ac
cording to a law which experiments have indicated, is there not a
necessary mathematical relation between this law and the general
fact of the equilibrium of heat ; and what is the physical cause of
this inequality in intensity ?

Lastly, when heat penetrates fluid masses, and determines in
them internal movements by continual changes of the temperature
and density of each molecule, can we still express, by differential

6 THEORY OF HEAT.

equations, the laws of such a compound effect ; and what is the
resulting change in the general equations of hydrodynamics ?

Such are the chief problems which I have solved, and which
have never yet been submitted to calculation. If we consider
further the manifold relations of this mathematical theory to
civil uses and the technical arts, we shall recognize completely
the extent of its applications. It is evident that it includes an
entire series of distinct phenomena, and that the study of it
cannot be omitted without losing a notable part of the science of
nature.

The principles of the theory are derived, as are those of
rational mechanics, from a very small number of primary facts,
the causes of which are not considered by geometers, but which
they admit as the results of common observations confirmed by all
experiment.

The differential equations of the propagation of heat express
the most general conditions, and reduce the physical questions to
problems of pure analysis, and this is the proper object of theory.
They are not less rigorously established than the general equations
of equilibrium and motion. In order to make this comparison
more perceptible, we have always preferred demonstrations ana
logous to those of the theorems which serve as the foundation
of statics and dynamics. These equations still exist, but receive
a different form, when they express the distribution of luminous
heat in transparent bodies, or the movements which the changes
of temperature and density occasion in the interior of fluids.
The coefficients which they contain are subject to variations whose
exact measure is not yet known ; but in all the natural problems
which it most concerns us to consider, the limits of temperature
differ so little that we may omit the variations of these co
efficients.

The equations of the movement of heat, like those which
express the vibrations of sonorous bodies, or the ultimate oscilla
tions of liquids, belong to one of the most recently discovered
branches of analysis, which it is very important to perfect. After
having established these differential equations their integrals must
be obtained ; this process consists in passing from a common
expression to a particular solution subject to all the given con
ditions. This difficult investigation requires a special analysis

PRELIMINARY DISCOURSE. 7

founded on new theorems, whose object we could not in this
place make known. The method which is derived from them
leaves nothing vague and indeterminate in the solutions, it leads
them up to the final numerical applications, a necessary condition
of every investigation, without which we should only arrive at
useless transformations.

The same theorems which have made known to us the
equations of the movement of heat, apply directly to certain pro
blems of general analysis a.nd dynamics whose solution has for a
long time been desired.

Profound study of nature is the most fertile source of mathe
matical discoveries. Not only has this study, in offering a de
terminate object to investigation, the advantage of excluding
vague questions and calculations without issue ; it is besides a
sure method of forming analysis itself, and of discovering the
elements which it concerns us to know, and which natural science
ought always to preserve : these are the fundamental elements
•which are reproduced in all natural effects.

We see, for example, that the same expression whose abstract
properties geometers had considered, and which in this respect
belongs to general analysis, represents as well the motion of light
in the atmosphere, as it determines the laws of diffusion of heat
in solid matter, and enters into all the chief problems of the
theory of probability.

The analytical equations, unknown to the ancient geometers,
which Descartes was the first to introduce into the study of curves
and surfaces, are not restricted to the properties of figures, and to
those properties which are the object of rational mechanics ; they
extend to all general phenomena. There cannot be a language
more universal and more simple, more free from errors and from
obscurities, that is to say more worthy to express the invariable
relations of natural things.

Considered from this point of view, mathematical analysis is as
extensive as nature itself; it defines all perceptible relations,
measures times, spaces, forces, temperatures ; this difficult science
is formed slowly, but it preserves every principle which it has once
acquired ; it grows and strengthens itself incessantly in the midst
of the many variations and errors of the human mind.

Its chief attribute is clearness ; it has no marks to express con-

8 THEORY OF HEAT.

fused notions. It brings together phenomena the most diverse,
and discovers the hidden analogies which unite them. If matter
escapes us, as that of air and light, by its extreme tenuity, if
bodies are placed far from us in the immensity of space, if man
wishes to know the aspect of the heavens at successive epochs
separated by a great number of centuries, if the actions of gravity
and of heat are exerted in the interior of the earth at depths
which will be always inaccessible, mathematical analysis can yet
lay hold of the laws of these phenomena. It makes them present
and measurable, and seems to be a faculty of the human mind
destined to supplement the shortness of life and the imperfec
tion of the senses ; and what is still more remarkable, it follows
the same course in the study of all phenomena ; it interprets them
by the same language, as if to attest the unity and simplicity of
the plan of the universe, and to make still more evident that
unchangeable order which presides over all natural causes.

The problems of the theory of heat present so many examples
of the simple and constant dispositions which spring from the
general laws of nature ; and if the order which is established in
these phenomena could be grasped by our senses, it would produce
in us an impression comparable to the sensation of musical sound.

The forms of bodies are infinitely varied ; the distribution of
the heat which penetrates them seems to be arbitrary and confused ;
but all the inequalities are rapidly cancelled and disappear as time
passes on. The progress of the phenomenon becomes more regular
and simpler, remains finally subject to a definite law which is the
same in all cases, and which bears no sensible impress of the initial
arrangement.

All observation confirms these consequences. The analysis
from which they are derived separates and expresses clearly, 1° the
general conditions, that is to say those which spring from the
natural properties of heat, 2° the effect, accidental but continued,
of the form or state of the surfaces ; 3° the effect, not permanent,
of the primitive distribution.

In this work we have demonstrated all the principles of the
theory of heat, and solved all the fundamental problems. They
could have been explained more concisely by omitting the simpler
problems, and presenting in the first instance the most general
results; but we wished to shew the actual origin of the theory and

PRELIMINARY DISCOURSE. 9

its gradual progress. When this knowledge has been acquired
and the principles thoroughly fixed, it is preferable to employ at
once the most extended analytical methods, as we have done in
the later investigations. This is also the course which we shall
hereafter follow in the memoirs which will be added to this work,
and which will form in some manner its complement *; and by this
means we shall have reconciled, so far as it can depend on our
selves, the necessary development of principles with the precision
which becomes the applications of analysis.

The subjects of these memoirs will be, the theory of radiant
heat, the problem of the terrestrial temperatures, that of the
temperature of dwellings, the comparison of theoretic results with
those which we have observed in different experiments, lastly the
demonstrations of the differential equations of the movement of
heat in fluids.

The work which we now publish has been written a long time
since ; different circumstances have delayed and often interrupted
the printing of it. In this interval, science has been enriched by
important observations ; the principles of our analysis, which had
not at first been grasped, have become better known ; the results
which we had deduced from them have been discussed and con
firmed. We ourselves have applied these principles to new
problems, and have changed the form of some of the proofs.
The delays of publication will have contributed to make the work
clearer and more complete.

The subject of our first analytical investigations on the transfer
of heat was its distribution amongst separated masses ; these have
been preserved in Chapter III., Section II. The problems relative
to continuous bodies, which form the theory rightly so called, were
solved many years afterwards ; this theory was explained for the
first time in a manuscript work forwarded to the Institute of
France at the end of the year 1807, an extract from which was
published in the Bulletin des Sciences (Societe Philomatique, year
1808, page 112). We added to this memoir, and successively for
warded very extensive notes, concerning the convergence of series,
the diffusion of heat in an infinite prism, its emission in spaces

1 These memoirs were never collectively published as a sequel or complement
to the Theorie Analytiquc de la Chaleur. But, as will be seen presently, the author
had written most of them before the publication of that work in 1822. [A. F.]

10 THEORY OF HEAT.

void of air, the constructions suitable for exhibiting the chief
theorems, and the analysis of the periodic movement at the sur
face of the earth. Our second memoir, on the propagation of
heat, was deposited in the archives of the Institute, on the 28th of
September, 1811. It was formed out of the preceding memoir and
the notes already sent in ; the geometrical constructions and
those details of analysis which had no necessary relation to the
physical problem were omitted, and to it was added the general
equation which expresses the state of the surface. This second
work was sent to press in the course of 1821, to be inserted in
the collection of the Academy of Sciences. It is printed without
any change or addition ; the text agrees literally with the deposited
manuscript, which forms part of the archives of the Institute \

In this memoir, and in the writings which preceded it, will be
found a first explanation of applications which our actual work

1 It appears as a memoir and supplement in volumes IV. and V. of the Me-
moircs de V Academic des Sciences. For convenience of comparison with, the table
of contents of the Analytical Theory of Heat, we subjoin the titles and heads of
the chapters of the printed memoir :

THEORIE DU MOUVEMENT DE LA CHALEUR DANS LES CORPS SOLIDES, PAR M.
FOURIER. [Memoires de V Academic Hoy ale des Sciences de Vlnstitut de France.
Tome IV. (for year 1819). Paris 1824.]

I. Exposition.

II. Notions generales et definitions preliminaires.

III. Equations du mouvement de la chaleur.

IV. Du mouvement lineaire et varie de la chaleur dans une armille.

V. De la propagation de la chaleur dans une lame rectangulaire dont Us temperatures
sont constantes.

VI. De la communication de la chaleur entre des masses disjointes.

VII. Du mouvement varie de la chaleur dans une sphere solide.

VIII. Du mouvement varie de la chaleur dans un cylindre solide.

IX. De la propagation de la chaleur dans un prisme dont Vextremite est assujcttie
a une temperature constante.

X. Du mouvement varie de la chaleur dans un solide de forme cubique.

XI. Du mouvement lineaire et varie de la chaleur dans les corps dont une dimension
est infinie.

SUITE DU MEMOIRS INTITULE: THEORIE DU MOUVEMENT DE LA CHALEUR DANS
LES CORPS SOLIDES; PAR M. FOURIER. [Memoires de V Academic Eoyale des Sciences
de rinstitut de France. Tome V. (for year 1820). Paris, 1826.]

XII. Des temperatures terrestres, et du mouvement de la chaleur dans Vinterieur
d'une sphere solide, dont la surface est assujettie a des changemens periodiques
de temperature.

XIII. Des lois mathematiques de Vequilibre de la chaleur rayonnante.

XIV. Comparaison des resultats de la theorie avec ceux de diverses experiences
[A. P.]

PRELIMINARY DISCOURSE. H

does not contain; they will be treated in the subsequent memoirs1
at greater length, and, if it be in our power, with greater clear
ness. The results of our labours concerning the same problems
are also indicated in several articles already published. The
extract inserted in the Annales de Chimie et de Physique shews
the aggregate of our researches (Vol. in. page 350, year 1816).
We published in the Annales two separate notes, concerning
radiant heat (Vol. iv. page 128, year 1817, and Vol. vi. page 259,
year 1817).

Several other articles of the same collection present the most
constant results of theory and observation ; the utility and the
extent of thermological knowledge could not be better appreciated
than by the celebrated editors of the Annales *.

In the Bulletin des Sciences (Societe philomatique year 1818,
page 1, and year 1820, page 60) will be found an extract from
a memoir on the constant or variable temperature of dwellings,
and an explanation of the chief consequences of our analysis of
the terrestrial temperatures.

M. Alexandre de Humboldt, whose researches embrace all the
great problems of natural philosophy, has considered the obser
vations of the temperatures proper to the different climates
from a novel and very important point of view (Memoir on Iso
thermal lines, Societe d'Arcueil, Vol. ill. page 462) ; (Memoir on
the inferior limit of perpetual snow, Annales de Chimie et de
Physique, Vol. v. page 102, year 1817).

As to the differential equations of the movement of heat in
fluids 3 mention has been made of them in the annual history of
the Academy of Sciences. The extract from our memoir shews
clearly its object and principle. (Analyse des travaux de VAca
demie des Sciences, by M. De Lambre, year 1820.)

The examination of the repulsive forces produced by heat,
which determine the statical properties of gases, does not belong

1 See note, page 9, and the notes, pages 11 — 13.

- Gay-Lussac and Arago. See note, p. 13.

3 Memoires de VAcademie des Sciences, Tome XII., Paris, 1833, contain on pp.
507—514, Me moire d'analyse sur le mouvement de la chaleur dans les fluides, par M.
Fourier. Lu a VAcademie Royale des Sciences, 4 Sep. 1820. It is followed on pp.
515 — 530 by Extrait des notes manuscrites conservees par Vavteur. The memoir
is signed Jh. Fourier, Paris, 1 Sep. 1820, but was published after the death of the
author. [A. F.]

12 THEORY OF HEAT.

to the analytical subject which, we have considered. This question
connected with the theory of radiant heat has just heen discussed
by the illustrious author of the Me'canique celeste, to whom all
the chief branches of mathematical analysis owe important
discoveries. (Connaissance des Temps, years 1824-5.)

The new theories explained in our work are united for ever
to the mathematical sciences, and rest like them on invariable
foundations ; all the elements which they at present possess they
will preserve, and will continually acquire greater extent. Instru
ments will be perfected and experiments multiplied. The analysis
which we have formed will be deduced from more general, that
is to say, more simple and more fertile methods common to many
classes of phenomena. For all substances, solid or liquid, for
vapours and permanent gases, determinations will be made of all
the specific qualities relating to heat, and of the variations of the
coefficients which express them1. At different stations on the
earth observations will be made, of the temperatures of the
ground at different depths, of the intensity of the solar heat and
its effects, constant or variable, in the atmosphere, in the ocean
and in lakes ; and the constant temperature of the heavens proper
to the planetary regions will become known2. The theory itself

1 Hemoires de VAcademie des Sciences, Tome VIII., Paris 1829, contain on
pp. 581 — 622, Memoire sur la Theorie Analytique de la Chaleur, par M. Fourier.
This was published whilst the author was Perpetual Secretary to the Academy.
The first only of four parts of the memoir is printed. The contents of all are
stated. I. Determines the temperature at any point of a prism whose terminal
temperatures are functions of the time, the initial temperature at any point being
a function of its distance from one end. II. Examines the chief consequences of
the general solution, and applies it to two distinct cases, according as the tempe
ratures of the ends of the heated prism are periodic or not. III. Is historical,
enumerates the earlier experimental and analytical researches of other writers
relative to the theory of heat ; considers the nature of the transcendental equations
appearing in the theory ; remarks on the employment of arbitrary functions ;
replies to the objections of M. Poisson ; adds some remarks on a problem of the
motion of waves. IV. Extends the application of the theory of heat by taking
account, in the analysis, of variations in the specific coefficients which measure
the capacity of substances for heat, the permeability of solids, and the penetra
bility of their surfaces. [A. F.]

2 Memoircs de VAcademie des Sciences, Tome VII. , Paris, 1827, contain on
pp. 569 — 604, Memoire sur les temperatures du globe terrestre et des espaces plane-
taires, par M. Fourier. The memoir is entirely descriptive ; it was read before the
Academy, 20 and 29 Sep. 1824 (Annales de Chimie et de Physique, 1824, xxvu.
p. 136). [A. F.]

PRELIMINARY DISCOURSE. 13

will direct all these measures, and assign their precision. No
considerable progress can hereafter be made which is not founded
on experiments such as these ; for mathematical analysis can
deduce from general and simple phenomena the expression of the
laws of nature ; but the special application of these laws to very
complex effects demands a long series of exact observations.

The complete list of the Articles on Heat, published by M. Fourier, in the
Annales de Chimie et de Physique, Series 2, is as follows :

1816. III. pp. 350—375. Theorie de la Chaleur (Extrait). Description by the
author of the 4to volume afterwards published in 1822 without the chapters on
radiant heat, solar heat as it affects the earth, the comparison of analysis with
experiment, and the history of the rise and progress of the theory of heat.

1817. IV. pp. 128 — 145. Note sur la Chaleur rayonnante. Mathematical
sketch on the sine law of emission of heat from a surface. Proves the author's
paradox on the hypothesis of equal intensity of emission in all directions.

1817. VI. pp. 259 — 303. Questions sur la theorie physique de la chaleur
rayonnante. An elegant physical treatise on the discoveries of Newton, Pictet,
Wells, TVollaston, Leslie and Prevost.

1820. XIII. pp. 418 — 438. Sur le refroidissement seculaire de la terre (Extrait).
Sketch of a memoir, mathematical and descriptive, on the waste of the earth's
initial heat.

1824. XXYII. pp. 136 — 167. Eemarques generates sur Ics temperatures du globe
terrestre et des espaces planetaires. This is the descriptive memoir referred to
above, Mem. Acad. d. Sc. Tome VII.

1824. XXYII. pp. 236 — 281. Eesume theorique des proprietes de la chaleur
rayonnante. Elementary analytical account of surface-emission and absorption
based on the principle of equilibrium of temperature.

1825. XXYIII. pp. 337 — 365. Eemarques sur la theorie mathematique de la
chaleur rayonnante. Elementary analysis of emission, absorption and reflection
by walls of enclosure uniformly heated. At p. 364, M. Fourier promises a Theorie
physique de la clialeur to contain the applications of the Theorie Analytique
omitted from the work published in 1822.

1828. XXXYII. pp. 291 — 315. Eecherches experimentales sur la faculte con-
ductrice des corps minces soumis a Vaction de la chaleur, et description d'un nouveau
thermometre de contact. A thermoscope of contact intended for lecture demonstra
tions is also described. M. Ernile Yerdet in his Conferences de Physique, Paris,
1872. Part I. p. 22, has stated the practical reasons against relying on the
theoretical indications of the thermometer of contact. [A. F.]

Of the three notices of memoirs by M. Fourier, contained in the Bulletin des
Sciences par la Societe Philomatique, and quoted here at pages 9 and 11, the first
was written by M. Poisson, the mathematical editor of the Bulletin, the other two by
M. Fourier. [A. F.]

THEORY OF HEAT.

Et ignem rcgunt numeri. — PLATO*.

CHAPTER I.

INTRODUCTION.

FIKST SECTION.

Statement of the Object of the Work.

1. THE effects of heat are subject to constant laws which
cannot be discovered without the aid of mathematical analysis.
The object of the theory which we are about to explain is to
demonstrate these laws ; it reduces all physical researches on
the propagation of heat, to problems of the integral calculus
whose elements are given by experiment. No subject has more
extensive relations with the progress of industry and the natural
sciences ; for the action of heat is always present, it penetrates
all bodies and spaces, it influences the processes of the arts,
and occurs in all the phenomena of the universe.

When heat is unequally distributed among the different parts
of a solid mass, it tends to attain equilibrium, and passes slowly
from the parts which are more heated to those which are less;
and at the same time it is dissipated at the surface, and lost
in the medium or in the void. The tendency to uniform dis
tribution and the spontaneous emission which acts at the surface
of bodies, change continually the temperature at their different
points. The problem of the propagation of heat consists in

1 Cf. Plato, Timaus, 53, B.

Sre 5' <?7rexeipetro KO a pel ad at TO Trav, trvp Trpwrov /cat yfjv Kal depa /cat vdup

[6 0eos] ddccrl re /cat dpiO/mois. [A. F.]

CH. I. SECT. I.] INTRODUCTION. 15

determining what is the temperature at each point of a body
at a given instant, supposing that the initial temperatures are
known. The following examples will more clearly make known
the nature of these problems.

2. If we expose to the continued and uniform action of a
source of heat, the same part of a metallic ring, whose diameter
is large, the molecules nearest to the source will be first heated,
and, after a certain time, every point of the solid will have
acquired very nearly the highest temperature which it can attain.
This limit or greatest temperature is not the same at different
points ; it becomes less and less according as they become more
distant from that point at which the source of heat is directly
applied.

When the temperatures have become permanent, the source
of heat supplies, at each instant, a quantity of heat which exactly
compensates for that which is dissipated at all the points of the
external surface of the ring.

If now the source be suppressed, heat will continue to be
propagated in the interior of the solid, but that which is lost
in the medium or the void, will no longer be compensated as
formerly by the supply from the source, so that all the tempe
ratures will vary and diminish incessantly until they have be
come equal to the temperatures of the surrounding medium.

3. Whilst the temperatures are permanent and the source
remains, if at every point of the mean circumference of the ring
an ordinate be raised perpendicular to the plane of the ring,
whose length is proportional to the fixed temperature at that
point, the curved line which passes through the ends of these
ordi nates will represent the permanent state of the temperatures,
and it is very easy to determine by analysis the nature of this
line. It is to be remarked that the thickness of the ring is
supposed to be sufficiently small for the temperature to be
sensibly equal at all points of the same section perpendicular
to the mean circumference. When the source is removed, the
line which bounds the ordinates proportional to the temperatures
at the different points will change its form continually. The
problem consists in expressing, by one equation, the variable

16 THEORY OF HEAT. [CHAP. I.

form of this curve, and in thus including in a single formula
all the successive states of the solid.

•

4. Let z be the constant temperature at a point m of the

mean circumference, x the distance of this point from the source,
that is to say the length of the arc of the mean circumference,
included between the point m and the point o which corresponds
to the position of the source; z is the highest temperature
which the point m can attain by virtue of the constant action
of the source, and this permanent temperature z_ isj*^ function
/(#) of the distance x. The first part of theC^roblemj consists
in determining the function f(x) which represents the permanent
state of the solid.

Consider next the variable state which succeeds to the former
state as soon as the source has been removed ; denote by t the
time which has passed since the suppression of the source, and
by v the value of the temperature at the point m after the
time t. The quantity v will be a certain function F (x, t) of
the distance x and the time t\ the object of the (pf oblem^is to
discover this function F (x, t), of which we only Imowas yet
that the initial value is f (x}, so that we ought to have the
equation f (.r) = F (x, o).

5. If we place a solid homogeneous mass, having the form
of a sphere or cube, in a medium maintained at a constant tem
perature, and if it remains immersed for a very long time, it will
acquire at all its points a temperature differing very little from
that of the fluid. Suppose the mass to be withdrawn in order
to transfer it to a cooler medium, heat will begin to be dissi
pated at its surface ; the temperatures at different points of the
mass will not be sensibly the same, and if we suppose it divided
into an infinity of layers by surfaces parallel to its external sur
face, each of those layers will transmit, at each instant, a certain
quantity of heat to the layer which surrounds it. If it be
imagined that each molecule carries a separate thermometer,
-which indicates its temperature at every instant, the state of
the solid will from time to time be represented by the variable
system of all these thermometric heights. It is required to
express the successive states by analytical formulae, so that we

SECT. I.] INTRODUCTION. 17

may know at any given instant the temperatures indicated by
each thermometer, and compare the quantities of heat which
flow during the same instant, between two adjacent layers, or
into the surrounding medium.

G. If the mass is spherical, and we denote by x the distance
of a point of this mass from the centre of the sphere, by t the
time which has elapsed since the commencement of the cooling,
and by v the variable temperature of the point m, it is easy to see
that all points situated at the same distance x from the centre
of the sphere have the same temperature v. This quantity v is a
certain function F (x, t} of the radius x and of the time t ; it must
be such that it becomes constant whatever be the value of x, when
we suppose t to be nothing ; for by hypothesis, the temperature at
all points is the same at the moment of emersion. The problem
consists in determining that function of x and t which expresses
the value of v.

7. In the next place it is to be remarked, that during the
cooling, a certain quantity of heat escapes, at each instant, through
the external surface, and passes into the medium. The value of
this quantity is not constant ; it is greatest at the beginning of the
cooling. If however we consider the variable state of the internal
spherical surface whose radius is x, we easily see that there must
be at each instant a certain quantity of heat which traverses that
surface, and passes through that part of the mass which is more
distant from the centre. This continuous flow of heat is variable
like that through the external surface, and both are quantities
comparable with each other ; their ratios are numbers whose vary
ing values are functions of the distance x, and of the time t which
has elapsed. It is required to determine these functions.

8. If the mass, which has been heated by a long immersion in
a medium, and whose rate of cooling we wish to calculate, is
of cubical form, and if we determine the position of each point mby
three rectangular co-ordinates x, y, z, taking for origin the centre
of the cube, and for axes lines perpendicular to the faces, we see
that the temperature v of the poiat m after the time t, is a func
tion of the four variables x, y, z, and t. The quantities of heat

F. H. 2

18 THEORY OF HEAT. [CHAP. I.

which flow out at each instant through the whole external surface
of the solid, are variable and comparable with each other ; their
ratios are analytical functions depending on the time t, the expres
sion of which must be assigned.

9. Let us examine also the case in which a rectangular prism
of sufficiently great thickness and of infinite length, being sub
mitted at its extremity to a constant temperature, whilst the air
which surrounds it is maintained at a less temperature, has at last
arrived at a fixed state which it is required to determine. All the
points of the extreme section at the base of the prism have, by
hypothesis, a common and permanent temperature. It is not the
same with a section distant from the source of heat; each of the
points of this rectangular surface parallel to the base has acquired
a fixed temperature, but this is not the same at different points of
the same section, and must be less at points nearer to the surface
exposed to the air. We see also that, at each instant, there flows
across a given section a certain quantity of heat, which always
remains the same, since the state of the solid has become constant.
The problem consists in determining the permanent temperature
at any given point of the solid, and the whole quantity of heat
which, in a definite time, flows across a section whose position is
given.

10. Take as origin of co-ordinates DC, y, z, the centre of the
base of the prism, and as rectangular axes, the axis of the prism
itself, and the two perpendiculars on the sides : the permanent
temperature v of the point m, whose co-ordinates are #, y, z, is
a function of three variables F (x, y, z) : it has by hypothesis a
constant value, when we suppose x nothing, whatever be the values
of y and z. Suppose we take for the unit of heat that quantity
which in the unit of time would emerge from an area equal to a
unit of surface, if the heated mass which that area bounds, and
which is formed of the same substance as the prism, were continu
ally maintained at the temperature of boiling water, and immersed
in atmospheric air maintained at the temperature of melting ice.

We see that the quantity of heat which, in the permanent
state of the rectangular prism, flows, during a unit of time, across
a certain section perpendicular to the axis, has a determinate ratio

SECT. I.] INTRODUCTION. 19

to the quantity of heat taken as unit. This ratio is not the same
for all sections : it is a function $ (#) of the distance r, at which
the section is situated. It is required to find an analytical expres
sion of the function <f> (#).

11. The foregoing examples suffice to give an exact idea of
the different problems which we have discussed.

The solution of these problems has made us understand that
the effects of the propagation of heat depend in the case of every
solid substance, on three elementary qualities, which are, its capa
city for heat, its own conducMity, and the exterior conducibility.

It has been observed that if two bodies of the same volume
and of different nature have equal temperatures, and if the same
quantity of heat be added to them, the increments of temperature
are not the same; the ratio of these increments is the, ratio of
their capacities for heat. In this manner, the first of the three
specific elements which regulate the action of heat is exactly
defined, and physicists have for a long time known several methods
of determining its value. It is not the same with the two others ;
their effects have often been observed, but there is but one exact
theory which can fairly distinguish, define, and measure them
with precision.

The proper or interior conducibility of a body expresses the
facility with which heat is propagated in passing from one internal
molecule to another. The external or relative conducibility of a
solid body depends on the facility with which heat penetrates the
surface, and passes from this body into a given medium, or passes
from the medium into the solid. The last property is modified by
the more or less polished state of the surface ; it varies also accord
ing to the medium in which the body is immersed ; but the
interior conducibility can change only with the nature of the
solid.

These three elementary qualities are represented in our
formulae by constant numbers, and the theory itself indicates
experiments suitable for measuring their values. As soon as they
are determined, all the problems relating to the propagation of
heat depend only on numerical analysis. The knowledge of these
specific properties may be directly useful in several applications of
the physical sciences ; it is besides an element in the study and

2—2

20 THEORY OF HEAT. [CHAP. I.

description of different substances. It is a very imperfect know
ledge of bodies which ignores the relations which they have with
one of the chief agents of nature. In general, there is no mathe
matical theory which has a closer relation than this with public
economy, since it serves to give clearness and perfection to the
practice of the numerous arts which are founded on the employ
ment of heat.

12. The problem of the terrestrial temperatures presents
one of the most beautiful applications of the theory of heat ; the
general idea to be formed of it is this. Different parts of the
surface of the globe are unequally exposed to the influence of the
solar rays; the intensity of their action depends on the latitude of
the place ; it changes also in the course of the day and in the
course of the year, and is subject to other less perceptible in
equalities. It is evident that, between the variable state of the
surface and that of the internal temperatures, a necessary relation
exists, which may be derived from theory. We know that, at a
certain depth below the surface of the earth, the temperature at a
given place experiences no annual variation: this permanent
underground temperature becomes less and less according as the
place is more and more distant from the equator. We may then
leave out of consideration the exterior envelope, the thickness of
which is incomparably small with respect to the earth's radius,
and regard our planet as a nearly spherical mass, whose surface
is subject to a temperature which remains constant at all points
on a given parallel, but is not the same on another parallel. It
follows from this that every internal molecule has also a fixed tem
perature determined by its position. The mathematical problem
consists in discovering the fixed temperature at any given point,
and the law which the solar heat follows whilst penetrating the
interior of the earth.

This diversity of temperature interests us still more, if we
consider the changes which succeed each other in the envelope
itself on the surface of which we dwell. Those alternations of
heat and cold which are reproduced everyday and in the course of
every year, have been up to the present time the object of repeated
observations. These we can now submit to calculation, and from
a common theory derive all the particular facts which experience

SECT. I.] INTRODUCTION. 21

has taught us. The problem is reducible to the hypothesis that
every point of a vast sphere is affected by periodic temperatures ;
analysis then tells us according to what law the intensity of these
variations decreases according as the depth increases, what is the
amount of the annual or diurnal changes at a given depth, the
epoch of the changes, and how the fixed value of the underground
temperature is deduced from the variable temperatures observed
at the surface.

13. The general equations of the propagation of heat are
partial differential equations, and though their form is very simple
the known methods l do not furnish any general mode of integrat
ing them; we could not therefore deduce from them the values
of the temperatures after a definite time. The numerical inter
pretation of the results of analysis is however necessary, and it
is a degree of perfection which it would be very important to give
to every application of analysis to the natural sciences. So long
as it is not obtained, the solutions may be said to remain in
complete and useless, and the truth which it is proposed to
discover is no less hidden in the formulas of analysis than it was
in the physical problem itself. We have applied ourselves with
much care to this purpose, and we have been able to overcome
the difficulty in all the problems of which we have treated, and
which contain the chief elements of the theory of heat. There is
not one of the problems whose solution does not provide conve
nient and exact means for discovering the numerical values of the
temperatures acquired, or those of the quantities of heat which

1 For the modern treatment of these equations consult

Partielle Differentialgleichungen, von B. Eiemann, Braunschweig, 2nd Ed., 1876.
The fourth section, Bewegung der Warme in festen Korpern.

Cours de physique mathematique, par E. Matthieu, Paris, 1873. The parts
relative to the differential equations of the theory of heat.

The Functions of Laplace, Lame, and Bessel, by I. Todhunter, London, 1875.
Chapters XXI. XXV.— XXIX. which give some of Lame's methods.

Conferences de Physique, par E. Verdet, Paris, 1872 [(Euvres, Vol. iv. Part i.].
Legons sur la propagation de la chaleur par conductibilite. These are followed by
a very extensive bibliography of the whole subject of conduction of heat.

For an interesting sketch and application of Fourier's Theory see

Theory of Heat, by Prof. Maxwell, London, 1875 [4th Edition]. Chapter XVIII.
On the diffusion of heat by conduction.

Natural Philosophy, by Sir W. Thomson and Prof. Tait, Vol. i. Oxford, 1867.
Chapter VII. Appendix D, On the secular cooling of the earth. [A. F. ]

22 THEORY OF HEAT. [CHAP. I.

have flowed through, when the values of the time and of the
variable coordinates are known. Thus will be given not only the
differential equations which the functions that express the values
of the temperatures must satisfy; but the functions themselves
will be given under a form which facilitates the numerical
applications.

14. In order that these solutions might be general, and have
an extent equal to that of the problem, it was requisite that they
should accord with the initial state of the temperatures, which is
arbitrary. The examination of this condition shews that we may
develop in convergent series, or express by definite integrals,
functions which are not subject to a constant law, and which
represent the ordinates of irregular or discontinuous lines. This
property throws a new light on the theory of partial differen
tial equations, and extends the employment of arbitrary functions
by submitting them to the ordinary processes of analysis.

15. It still remained to compare the facts with theory. With
this view, varied and exact experiments were undertaken, whose
results were in conformity with those of analysis, and gave them
an authority which one would have been disposed to refuse to
them in a new matter which seemed subject to so much uncer
tainty. These experiments confirm the principle from which we
started, and which is adopted by all physicists in spite of the
diversity of their hypotheses on the nature of heat.

16. Equilibrium of temperature is effected not only by way
of contact, it is established also between bodies separated from
each other, which are situated for a long time in the same region.
This effect is independent of contact with a medium; we have
observed it in spaces wholly void of air. To complete our theory
it was necessary to examine the laws which radiant heat follows,
on leaving the surface of a body. It results from the observations
of many physicists and from our own experiments, that the inten
sities of the different rays, which escape in all directions from any
point in the surface of a heated body, depend on the angles which
their directions make with the surface at the same point. We
have proved that the intensity of a ray diminishes as the ray

SECT. I.] INTRODUCTION. 23

makes a smaller angle with the element of surface, and that it is
proportional to the sine of that angle \ This general law of
emission of heat which different observations had already indi
cated, is a necessary consequence of the principle of the equilibrium
of temperature and of the laws of propagation of heat in solid
bodies.

Such are the chief problems which have been discussed in
this work; they are all directed to one object only, that is to
establish clearly the mathematical principles of the theory of heat,
and to keep up in this way with the progress of the useful arts,
and of the study of nature.

17. From what precedes it is evident that a very extensive
class of phenomena exists, not produced by mechanical forces, but
resulting simply from the presence and accumulation of heat.
This part of natural philosophy cannot be connected with dy
namical theories, it has principles peculiar to itself, and is founded
on a method similar to that of other exact sciences. The solar
heat, for example, which penetrates the interior of the globe, dis
tributes itself therein according to a regular law which does not
depend on the laws of motion, and cannot be determined by the
principles of mechanics. The dilatations which the repulsive
force of heat produces, observation of which serves to measure
temperatures, are in truth dynamical effects; but it is not these
dilatations which we calculate, when we investigate the laws of
the propagation of heat.

18. There are other more complex natural effects, which
depend at the same time on the influence of heat, and of attrac
tive forces: thus, the variations of temperatures which the move
ments of the sun occasion in the atmosphere and in the ocean,
change continually the density of the different parts of the air
and the waters. The effect of the forces which these masses obey
is modified at every instant by a new distribution of heat, and
it cannot be doubted that this cause produces the regular winds,
and the chief currents of the sea; the solar and lunar attractions
occasioning in the atmosphere effects but slightly sensible, and
not general displacements. It was therefore necessary, in order to

1 Mem. Acad. d. Sc. Tome V. Paris, 1826, pp. 179—213. [A. F.]

24 THEORY OF HEAT. [CHAP. I.

submit these grand phenomena to calculation, to discover the
mathematical laws of the propagation of heat in the interior of
masses.

19. It will be perceived, on reading this work, that heat at
tains in bodies a regular disposition independent of the original
distribution, which may be regarded as arbitrary.

In whatever manner the heat was at first distributed, the
system of temperatures altering more and more, tends to coincide
sensibly with a definite state which depends only on the form of
the solid. In the ultimate state the temperatures of all the points
are lowered in the same time, but preserve amongst each other the
same ratios : in order to express this property the analytical for
mulae contain terms composed of exponentials and of quantities
analogous to trigonometric functions.

Several problems of mechanics present analogous results, such as
the isochronism of oscillations, the multiple resonance of sonorous
bodies. Common experiments had made these results remarked,
and analysis afterwards demonstrated their true cause. As to
those results which depend on changes of temperature, they could
not have been recognised except by very exact experiments ; but
mathematical analysis has outrun observation, it has supplemented
our senses, and has made us in a manner witnesses of regular and
harmonic vibrations in the interior of bodies.

20. These considerations present a singular example of the
relations which exist between the abstract science of numbers
and natural causes.

When a metal bar is exposed at one end to the constant action
of a source of heat, and every point of it has attained its highest
temperature, the system of fixed temperatures corresponds exactly
to a table of logarithms ; the numbers are the elevations of ther
mometers placed at the different points, and the logarithms are
the distances of these points from the source. In general heat
distributes itself in the interior of solids according to a simple law
expressed by a partial differential equation common to physical
problems of different order. The irradiation of heat has an evident
relation to the tables of sines, for the rays which depart from the
same point of a heated surface, differ very much from each other,

SECT. I.] INTRODUCTION. 25

and their intensity is rigorously proportional to the sine of the
angle which the direction of each ray makes with the element of
surface.

If we could observe the changes of temperature for every in
stant at every point of a solid homogeneous mass, we should dis
cover in these series of observations the properties of recurring
series, as of sines and logarithms ; they would be noticed for
example in the diurnal or annual variations of temperature of
different points of the earth near its surface.

We should recognise again the same results and all the chief
elements of general analysis in the vibrations of elastic media, in
the properties of lines or of curved surfaces, in the movements of
the stars, and those of light or of fluids. Thus the functions ob
tained by successive differentiations, which are employed in the
development of infinite series and in the solution of numerical
equations, correspond also to physical properties. The first of
these functions, or the fluxion properly so called, expresses in
geometry the inclination of the tangent of a curved line, and in
dynamics the velocity of a moving body when the motion varies ; '
in the theory of heat it measures the quantity of heat which flows
at each point of a body across a given surface. Mathematical
analysis has therefore necessary relations with sensible phenomena ;
its object is not created by human intelligence; it is a pre-existent
element of the universal order, and is not in any way contingent
or fortuitous ; it is imprinted throughout all nature.

21. Observations more exact and more varied will presently
ascertain whether the effects of heat are modified by causes which
have not yet been perceived, and the theory will acquire fresh
perfection by the continued comparison of its results with the
results of experiment ; it will explain some important phenomena
which we have not yet been able to submit to calculation ; it will
shew how to determine all the therm ornetric effects of the solar
rays, the fixed or variable temperature which would be observed at
different distances from the equator, whether in the interior of
the earth or beyond the limits of the atmosphere, whether in the
ocean or in different regions of the air. From it will be derived
the mathematical knowledge of the great movements which result
from the influence of heat combined with that of gravity. The

26 THEORY OF HEAT. [CHAP. I.

same principles will serve to measure the conducibilities, proper or
relative, of different bodies, and their specific capacities, to dis
tinguish all the causes which modify the emission of heat at the
surface of solids, and to perfect thermometric instruments.

The theory of heat will always attract the attention of ma
thematicians, by the rigorous exactness of its elements and the
analytical difficulties peculiar to it, and above all by the extent
and usefulness of its applications ; for all its consequences con
cern at the same time general physics, the operations of the arts,
domestic uses and civil economy.

SECTION II.

Preliminary definitions and general notions.

22. OF the nature of heat uncertain hypotheses only could be
formed, but the knowledge of the mathematical laws to which its
effects are subject is independent of all hypothesis ; it requires only
an attentive examination of the chief facts which common obser
vations have indicated, and which have been confirmed by exact
experiments.

It is necessary then to set forth, in the first place, the general
results of observation, to give exact definitions of all the elements
of the analysis, and to establish the principles upon which this
analysis ought to be founded.

The action of heat tends to expand all bodies, solid, liquid or
gaseous ; this is the property which gives evidence of its presence.
Solids and liquids increase in volume^ if the quantity of heat which
they contain increases ; they contract if it diminishes.

When all the parts of a solid homogeneous body, for example
those of a mass of metal, are equally heated, and preserve without
any change the same quantity of heat, they have also and retain
the same density. This state is expressed by saying that through
out the whole extent of the mass the molecules have a common
and permanent temperature.

23. The thermometer is a body whose smallest changes of
volume can be appreciated ; it serves to measure temperatures by

SECT. II.] PRELIMINARY DEFINITIONS. 27

the dilatation of a fluid or of air. We assume the construction,
use and properties of this instrument to be accurately known.
The temperature of a body equally heated in every part, and
which keeps its heat, is that which the thermometer indicates
when it is and remains in perfect contact with the body in
question.

Perfect contact is when the thermometer is completely im
mersed in a fluid mass, and, in general, when there is no point of
the external surface of the instrument which is not touched by one
of the points of the solid or liquid mass whose temperature is to be
measured. In experiments it is not always necessary that this con
dition should be rigorously observed ; but it ought to be assumed
in order to make the definition exact.

24. Two fixed temperatures are determined on, namely : the
temperature of melting ice which is denoted by 0, and the tern-
perature of boiling water which we will denote by 1 : the water is
supposed to be boiling under an atmospheric pressure represented
by a certain height of the barometer (76 centimetres), the mercury
of the barometer being at the temperature 0.

25. Different quantities of heat are measured by determining
how many times they contain a fixed quantity which is taken as
the unit. Suppose a mass of ice having a definite weight (a kilo
gramme) to be at temperature 0, and to be converted into water at
the same temperature 0 by the addition of a certain quantity of
heat : the quantity of heat thus added is taken as the unit of
measure. Hence the quantity of heat expressed by a number C
contains C times the quantity required to diaoolvo a kilogramme
of ice at the temperature zero into a mass of water at the same
zero temperature.

26. To raise a metallic mass having a certain weight, a kilo
gramme of iron for example, from the temperature 0 to the
temperature 1, a new quantity of heat must be added to that
which is already contained in the mass. The number C which
denotes this additional quantity of heat, is the specific capacity of
iron for heat; the number C has very different values for different
substances.

28 THEORY OF HEAT. [CHAP. I.

27. If a body of definite nature and weight (a kilogramme of
mercury) occupies a volume Fat temperature 0, it will oecupy a
greater volume F+ A, when it has acquired the temperature 1,
that is to say, when the heat which it contained at the tempera
ture 0 has been increased by a new quantity C, equal to the
specific capacity of the body for heat. But if, instead of adding
this quantity C, a quantity z C is added (z being a number
positive or negative) the new volume will be F + B instead
of F + A. Now experiments shew that if | is equal to J, the
increase of volume 8 is only half the total increment A, and
that in general the value of B is ^A, when the quantity of heat
added is zC.

28. The ratio z of the two quantities zG and C of heat added,
which is the same as the ratio of the two increments of volume 8
and A, is that which is called the temperature; hence the quantity
which expresses the actual temperature of a body represents the
excess of its actual volume over the volume which it would occupy
at the temperature of melting ice, unity representing the whole
excess of volume which corresponds to the boiling point of
water, over the volume which corresponds to the melting point
of ice.

29. The increments of volume of bodies are in general pro
portional to the increments of the quantities of heat which
produce the dilatations, but it must be remarked that this propor
tion is exact only in the case where the bodies in question are
subjected to temperatures remote from those which determine
their change of state. The application of these results to all
liquids must not be relied on; and with respect to water in
particular, dilatations do not always follow augmentations of
heat.

In general the temperatures are numbers proportional to the
quantities of heat added, and in the cases considered by us,
these numbers are proportional also to the increments of
volume.

30. Suppose that a body bounded by a plane surface having
a certain area (a square metre) is maintained in any manner

SECT. II.] PRELIMINARY DEFINITIONS. 29

•whatever at constant temperature 1, common to all its points,
and that the surface in question is in contact with air maintained
at temperature 0 : the heat which escapes continuously at the
surface and passes into the surrounding medium will be replaced
always by the heat which proceeds from the constant cause to
whose action the body is exposed; thus, a certain quantity of heat
denoted by h will flow through the surface in a definite time (a
minute).

This amount_ ^ of a flow continuous and always similar to
itself, which takes place at a unit of surface at a fixed temperature,
is the measure of the external conducibility of the body, that is
to say, of the facility with which its surface transmits heat to the
atmospheric air.

The air is supposed to be continually displaced with a given
uniform velocity : but if the velocity of the current increased, the
quantity of heat communicated to the medium would vary also :
the same would happen if the density of the medium were

iucrease ~

31. If the excess of the constant temperature of the body
over the temperature of surrounding bodies, instead of being equal
to 1, as has been supposed, had a less value, the quantity of heat
dissipated would be less than k. The result of observation is,
as we shall see presently, that this quantity of heat lost may be
regarded as sensibly proportional to the excess of the temperature
of the body over that of the air and surrounding bodies. Hence
the quantity h having been determined by one experiment in
which the surface heated is at temperature 1, and the medium at
temperature 0; we conclude that hz would be the quantity, if the
temperature of the surface were z, all the other circumstances
remaining the same. This result must be admitted when z is a
small fraction.

32. The value h of the quantity of heat which is dispersed
across a heated surface is different for different bodies; and it
varies for the same body according to the different states of the
surface. The effect of irradiation diminishes as the surface
becomes more polished; so that by destroying the polish of the
surface the value of h is considerably increased. A heated

30 THEORY OF HEAT. [CHAP. I.

metallic body will be more quickly cooled if its external surface is
covered with a black coating such as will entirely tarnish its
metallic lustre.

33. The rays of heat which escape from the surface of a body
pass freely through spaces void of air; they are propagated also
in atmospheric air: their directions are not disturbed by agitations
in the intervening air: they can be reflected by metal mirrors
and collected at their foci. Bodies at a high temperature, when
plunged into a liquid, heat directly only those parts of the mass
with which their surface is in contact. The molecules whose dis
tance from this surface is not extremely small, receive no direct
heat; it is not the same with aeriform fluids; in these the rays of
heat are borne with extreme rapidity to considerable distances,
whether it be that part of these rays traverses freely the layers of
air, or whether these layers transmit the rays suddenly without
altering their direction.

34. When the heated body is placed in air which is main
tained at a sensibly constant temperature, the heat communicated
to the air makes the layer of the fluid nearest to the surface of the
body lighter; this layer rises more quickly the more intensely it is
heated, and is replaced by another mass of cool air. A current
is thus established in the air whose direction is vertical, and
whose velocity is greater as the temperature of the body is higher.
For this reason if the body cooled itself gradually the velocity of
the current would diminish with the temperature, and the law
of cooling would not be exactly the same as if the body were
exposed to a current of air at a constant velocity.

35. When bodies are sufficiently heated to diffuse a vivid light,
part of their radiant heat mixed with that light can traverse trans
parent solids or liquids, and is subject to the force which produces
refraction. The quantity of heat which possesses this faculty
becomes less as the bodies are less inflamed ; it is, we may say,
insensiblefor very opaque bodies however highly theymaybe heated.
A thin transparent plate intercepts almost all the direct heat
which proceeds from an ardent mass of metal ; but it becomes
heated in proportion as the intercepted rays are accumulated in

SECT. II.] PRELIMINARY DEFINITIONS. 31

it ; whence, if it is formed of ice, it becomes liquid ; but if this
plate of ice is exposed to the rays of a torch it allows a sensible
amount of heat to pass through with the light.

36. We have taken as the measure of the external conduci-
bility of a solid body a coefficient h, which denotes the quantity of
heat which would pass, in a definite time (a minute), from the
surface of this body, into atmospheric air, supposing that the sur
face had a definite extent (a square metre), that the constant
temperature of the body was 1, and that of the air 0, and that
the heated surface was exposed to a current of air of a given in
variable velocity. This value of h is determined by observation.
The quantity of heat expressed by the coefficient is composed of
two distinct parts which cannot be measured except by very exact
experiments. One is the heat communicated by way of contact to
the surrounding air : the other, much less than the first, is the
radiant heat emitted. We must assume, in our first investigations,
that the quantity of heat lost does not change when the tempera
tures of the body and of the medium are augmented by the same
sufficiently small quantity.

37. Solid substances differ again, as we have already remarked,
by their property of being more or less permeable to heat ; this
quality is their conducibility proper: we shall give its definition and
exact measure, after having treated of the uniform and linear pro
pagation of heat. Liquid substances possess also the property of
transmitting heat from molecule to molecule, and the numerical
value of their conducibility varies according to the nature of the
substances : but this effect is observed with difficulty in liquids,
since their molecules change places on change of temperature. The
propagation q£ heat in them depends chiefly on this continual dis
placement, in all cases where the lower parts of the mass are most
exposed to the action of the source of heat. If, on the contrary,
the source of heat be applied to that part of the mass which is
highest, as was the case in several of our experiments, the transfer
of heat, which is very slow, does not produce any displacement,
at least when the increase of temperature does not diminish the
volume, as is indeed noticed in singular cases bordering on changes
of state.

32 THEORY OF HEAT. [CHAP. I.

38. To this explanation of the chief results of observation, a
general remark must be added on equilibrium of temperatures;
which consists in this, that different bodies placed in the same re
gion, all of whose parts are and remain equally heated, acquire also
a common 'and permanent temperature.

Suppose that all the parts of a mass M have a common and
constant temperature a, which is maintained by any cause what
ever: if a smaller body m be placed in perfect contact with the
mass M, it will assume the common temperature a.

In reality this result would not strictly occur except after an
infinite time : but the exact meaning of the proposition is that if
the body m had the temperature a before being placed in contact,
it would keep it without any change. The same would be the case
with a multitude of other bodies n, p, q, r each of which was
placed separately in perfect contact with the mass M : all would
acquire the constant temperature a. Thus a thermometer if suc
cessively applied to the different bodies m, n,p, q, r would indicate
the same temperature.

39. The effect in question is independent of contact, and
would still occur, if every part of the body m were enclosed in
the solid M, as in an enclosure, without touching any of its parts.
For example, if the solid were a spherical envelope of a certain
thickness, maintained by some external cause at a temperature a,
and containing a space entirely deprived of air, and if the body m
could be placed in any part whatever of this spherical space, with
out touching any point of the internal surface of the enclosure, it
would acquire the common temperature a, or rather, it would pre
serve it if it had it already. The result would be the same for
all the other bodies n, p, q, r, whether they were placed separately
or all together in the same enclosure, and whatever also their sub
stance and form might be.

40. Of all modes of presenting to ourselves the action of
heat, that which seems simplest and most conformable to observa
tion, consists in comparing this action to that of light. Mole
cules separated from one another reciprocally communicate, across
empty space, their rays of heat, just as shining bodies transmit
their light.

SECT. II.] GENERAL NOTIONS. 33

If within an enclosure closed in all directions, and maintained
by some external cause at a fixed temperature a, we suppose dif
ferent bodies to be placed without touching any part of the bound
ary, different effects will be observed according as the bodies,
introduced into this space free from air, are more or less heated.
If, in the first instance, we insert only one of these bodies, at the
same temperature as the enclosure, it will send from all points of
its surface as much heat as it receives from the solid which sur
rounds it, and is maintained in its original state by this exchange
of equal quantities.

If we insert a second body whose temperature 6 is less than a,
it will at first receive from the surfaces which surround it on
all sides without touching it, a quantity of heat greater than that
which it gives out : it will be heated more and more and will
absorb through its surface more heat than in the first instance.

The initial temperature b continually rising, will approach with
out ceasing the fixed temperature «, so that after a certain time
the difference will be almost insensible. The effect would be op
posite if we placed within the same enclosure a third body whose
temperature was greater than a.

41. All bodies have the property of emitting heat through
their surface; the hotter they are the more they emit; the
intensity of the emitted rays changes very considerably with the
state of the surface.

42. Every surface which receives rays of heat from surround
ing bodies reflects part and admits the rest : the heat which is not
reflected, but introduced through the surface, accumulates within
the solid; and so long as it exceeds the quantity dissipated by
irradiation, the temperature rises.

43. The rays which tend to go out of heated bodies are
arrested at the surface by a force which reflects part of them into
the interior of the mass. The cause which hinders the incident
rays from traversing the surface, and which divides these rays into
two parts, of which one is reflected and the other admitted, acts in
the same manner on the rays which are directed from the interior
of the body towards external space.

F. H. 3

34 THEORY OF HEAT. [CHAP. I.

If by modifying the state of the surface we increase the force
by which it reflects the incident rays, we increase at the same time
the power which it has of reflecting towards the interior of the
body rays which are tending to go out. The incident rays intro
duced into the mass, and the rays emitted through the surface, are
equally diminished in quantity.

44. If within the enclosure above mentioned a number of
bodies were placed at the same time, separate from each other
and unequally heated, they would receive and transmit rays of heat
so that at each exchange their temperatures would continually
vary, and would all tend to become equal to the fixed temperature
of the enclosure.

This effect is precisely the same as that which occurs when
heat is propagated within solid bodies ; for the molecules which
compose these bodies are separated by spaces void of air, and
have the property of receiving, accumulating and emitting heat.
Each of them sends out rays on all sides, and at the same time
receives other rays from the molecules which surround it.

* 45. The heat given out by a point situated in the interior of
a solid mass can pass directly to an extremely small distance only;
it is, we may say, intercepted by the nearest particles ; these parti
cles only receive the heat directly and act on more distant points.
It is different with gaseous fluids ; the direct effects of radiation
become sensible in them at very considerable distances.

46. Thus the heat which escapes in all directions from a part
of the surface of a solid, passes on in air to very distant points ; but
is emitted only by those molecules of the body which are extremely
near the surface. A point of a heated mass situated at a very
small distance from the plan^ superficies which separates the mass
from external space, sends to that space an infinity of rays, but
they do not all arrive there; they are diminished by all that quan
tity of heat which is arrested by the intermediate molecules of the
solid. The part of the ray actually dispersed into space becomes
less according as it traverses a longer path within the mass. Thus
the ray which escapes perpendicular to the surface has greater in
tensity than that which, departing from the same point, follows

SECT. II.] GENERAL NOTIONS. 35

an oblique direction, and the most oblique rays are wholly inter
cepted.

The same consequences apply to all the points which are near
enough to the surface to take part in the emission of heat, from
which it necessarily follows that the whole quantity of heat which
escapes from the surface in the normal direction is very much
greater than that whose direction is oblique. We have submitted
this question to calculation, and our analysis proves that the in
tensity of the ray is proportional to the sine of the angle which
the ray makes with the element of surface. Experiments had
already indicated a similar result.

47. This theorem expresses a general law which has a neces
sary connection with the equilibrium and mode of action of heat.
If the rays which escape from a heated surface had the same in
tensity in all directions, a thermometer placed at one of the points
of a space bounded on all sides by an enclosure maintained at a
constant temperature would indicate a temperature incomparably
greater than -that of the enclosure1. Bodies placed within this
enclosure would not take a common temperature, as is always
noticed; the temperature acquired by them would depend on the
place which they occupied, or on their form, or on the forms of
neighbouring bodies.

The same results would be observed, or other effects equally
opposed to common experience, if between the rays which escape
from the same point any other relations were admitted different
from those which we have enunciated. We have recognised this
law as the only one compatible with the general fact of the equi
librium of radiant heat.

48. If a space free from air is bounded on all sides by a solid
enclosure whose parts are maintained at a common and constant
temperature a, and if a thermometer, having the actual tempera
ture a, is placed at any point whatever of the space, its temperature
will continue without any change. It will receive therefore at
each instant from the inner surface of the enclosure as much heat
as it gives out to it. This effect of the rays of heat in a given
space is, properly speaking, the measure of the temperature : but

1 See proof by M. Fourier, Ann. d. Cli. et Ph. Ser. 2, iv. p. 128. [A. F.]

3—2

36 THEORY OF HEAT. [CHAP. I.

this consideration presupposes the mathematical theory of radiant
heat.

If now between the thermometer and a part of the surface of
the enclosure a body M be placed whose temperature is a, the
thermometer will cease to receive rays from one part of the inner
surface, but the rays will be replaced by those which it will re
ceive from the interposed body M. An easy calculation proves
that the compensation is exact, so that the state of the thermo
meter will be unchanged. It is not the same if the temperature
of the body M is different from that of the enclosure. When
it is greater, the rays which the interposed body M sends to the
thermometer and which replace the intercepted rays convey more
heat than the latter; the temperature of the thermometer must
therefore rise.

If, on the contrary, the intervening body has a temperature
less than a, that of the thermometer must fall; for the rays which
this body intercepts are replaced by those which it gives out, that
is to say, by rays cooler than those of the enclosure; thus the
thermometer does not receive all the heat necessary to maintain
its temperature a.

49. Up to this point abstraction has been made of the power
which all surfaces have of reflecting part of the rays wrhich are
sent to them. If this property were disregarded we should have
only a very incomplete idea of the equilibrium of radiant heat.

Suppose then that on the inner surface of the enclosure, main
tained at a constant temperature, there is a portion which enjoys,
in a certain degree, the power in question ; each point of the re
flecting surface will send into space two kinds of rays ; the one go
out from the very interior of the substance of which the enclosure is
formed, the others are merely reflected by the same surface against
which they had been sent. But at the same time that the surface
repels on the outside part of the incident rays, it retains in the
inside part of its own rays. In this respect an exact compensation
is established, that is to say, every one of its own rays which the
surface hinders from going out is replaced by a reflected ray of
equal intensity.

The same result would happen, if the power of reflecting rays
affected in any degree whatever other parts of the enclosure, or the

. II.] GENERAL NOTIONS. 37

surface of bodies placed within the same space and already at
the common temperature.

Thus the reflection of heat does not disturb the equilibrium
of temperatures, and does not introduce, whilst that equilibrium
exists, any change in the law according to which the intensity of
rays which leave the same point decreases proportionally to the
sine of the angle of emission.

50. Suppose that in the same enclosure, all of whose parts
maintain the temperature a, we place an isolated body M, and
a polished metal surface R, which, turning its concavity towards
the body, reflects great part of the rays which it received from the
body; if we place a thermometer between the body IT and the re
flecting surface R, at the focus of this mirror, three different effects
will be observed according as the temperature of the body J/ is
equal to the common temperature a, or is greater or less.

In the first case, the thermometer preserves the temperature
a ; it receives 1°, rays of heat from all parts of the enclosure not
hidden from" it by the body M or by the mirror ; 2°, rays given out
by the body ; 3°, those which the surface R sends out to the focus,
whether they come from the mass of the mirror itself, or whether its
surface has simply reflected them ; and amongst the last we may
distinguish between those which have been sent to the mirror by
the mass J/, and those which it has received from the enclosure.
All the rays in question proceed from surfaces which, by hypo
thesis, have a common temperature a, so that the thermometer
is precisely in the same state as if the space bounded by the en
closure contained 110 other body but itself.

In the second case, the thermometer placed between the heated
body M and the mirror, must acquire a temperature greater than
a. In reality, it receives the same rays as in the first hypothesis ;
but with two remarkable differences : one arises from the fact that
the rays sent by the body J/ to the mirror, and reflected upon the
thermometer, contain more heat than in the first case. The other
difference depends on the fact that the rays sent directly by the
body M to the thermometer contain more heat than formerly.
Both causes, and chiefly the first, assist in raising the tempera
ture of the thermometer.

In the third case, that is to say, when the temperature of the

38 THEORY OF HEAT. [CHAP. I.

mass M is less than a, the temperature must assume also a tem
perature less than a. In fact, it receives again all the varieties of
rays which we distinguished in the first case : but there are two
kinds of them which contain less heat than in this first hypothesis,
that is to say, those which, being sent out by the body M, are
reflected by the' mirror upon the thermometer, and those which
the same body M sends to it directly. Thus the thermometer floes
not receive all the heat which it requires to preserve its original
temperature a. It gives out more heat than it receives. It is
inevitable then that its temperature must fall to the point at
which the rays which it receives suffice to compensate those which
it loses. This last effect is what is called the reflection of cold,
and which, properly speaking, consists in the reflection of too
feeble heat. The mirror intercepts a certain quantity of heat, and
replaces it by a less quantity.

51. If in the enclosure, maintained at a constant temperature
a, a body M be placed, whose temperature a is less than a, the
presence of this body will lower the thermometer exposed to its
rays, and we may remark that the rays sent to the thermometer
from the surface of the body M, are in general of two kinds,
namely, those which come from inside the mass M, and those
which, coming from different parts of the enclosure, meet the sur
face M and are reflected upon the thermometer. The latter rays
have the common temperature a, but those which belong to the
body M contain less heat, and these are the rays which cool the
thermometer. If now, by changing the state of the surface of the
body M, for example, by destroying the polish, we diminish the
power which it has of reflecting the incident rays, the thermo
meter will fall still lower, and will assume a temperature a" less
than a. In fact all the conditions would be the same as in the
preceding case, if it were not that the body M gives out a greater
quantity of its own rays and reflects a less quantity of the rays-
which it receives from the enclosure; that is to say, these last rays,
which have the common temperature, are in part replaced by
cooler rays. Hence the thermometer no longer receives so much
heat as formerly.

If, independently of the change in the surface of the body M,
we place a metal mirror adapted to reflect upon the thermometer

SECT. II.] GENERAL NOTIONS. 39

the rays which have left M, the temperature will assume a value
a" less than a". The mirror, in fact, intercepts from the thermo
meter part of the rays of the enclosure which all have the tem
perature a, and replaces them by three kinds of rays ; namely,
1°, those which come from the interior of the mirror itself, and
which have the common temperature ; 2°, those which the different
parts of the enclosure send to the mirror with the same tempera
ture, and which are reflected to the focus ; 3°, those which, coming
from the interior of the body J/, fall upon the mirror, and are
reflected upon the thermometer. The last rays have a tempera
ture less than a ; hence the thermometer no longer receives so
much heat as it received before the mirror was set up.

Lastly, if we proceed to change also the state of the surface of
the mirror, and by giving it a more perfect polish, increase its
power of reflecting heat, the thermometer will fall still lower. In
fact, all the conditions exist which occurred in the preceding case.
Only, it happens that the mirror gives out a less quantity of its
own rays, and replaces them by those which it reflects. Now,
amongst these last rays, all those which proceed from the interior
of the mass M are less intense than if they had come from the
interior of the metal mirror ; hence the thermometer receives still
less heat than formerly : it will assume therefore a temperature
a"" less than a"'.

By the same principles all the known facts of the radiation of
heat or of cold are easily explained.

52. The effects of heat can by no means be compared with
those of an elastic fluid whose molecules are at rest.

It would be useless to attempt to deduce from this hypothesis
the laws of propagation which we have explained in this work,
and which all experience has confirmed. The free state of heat is
the same as that of light ; the active state of this element is then
entirely different from that of gaseous substances. Heat acts in
the same manner in a vacuum, in elastic fluids, and in liquid or
solid masses, it is propagated only by way of radiation, but its
sensible effects differ according to the nature of bodies.

53. Heat is the origin of all elasticity ; it is the repulsive
force which preserves the form of solid masses, and the volume of

40 THEORY OF HEAT. [CHAP. I.

liquids. In solid masses, neighbouring molecules would yield to
their mutual attraction, if its effect were not destroyed by the
heat which separates them.

This elastic force is greater according as the temperature is
higher ; which is the reason why bodies dilate or contract when
their temperature is raised or lowered.

54 The equilibrium which exists, in the interior of a solid
mass, between the repulsive force of heat and the molecular attrac
tion, is stable ; that is to say, it re-establishes itself when disturbed
by an accidental cause. If the molecules are arranged at distances
proper for equilibrium, and if an external force begins to increase
this distance without any change of temperature, the effect of
attraction begins by surpassing that of heat, and brings back the
molecules to their original position, after a multitude of oscillations
which become less and less sensible.

A similar effect is exerted in the opposite sense when a me
chanical cause diminishes the primitive distance of the molecules ;
such is the origin of the vibrations of sonorous or flexible bodies,
and of all the effects of their elasticity.

55. In the liquid or gaseous state of matter, the external
pressure is additional or supplementary to the molecular attrac
tion, and, acting on the surface, does not oppose change of form,
but only change of the volume occupied. Analytical investigation
will best shew how the repulsive force of heat, opposed to the
attraction of the molecules or to the external pressure, assists in
the composition of bodies, solid or liquid, formed of one or more
elements, and determines the elastic properties of gaseous fluids ;
but these researches do not belong to the object before us, and
appear in dynamic theories.

56. It cannot be doubted that the mode of action of heat
always consists, like that of light, in the reciprocal communication
of rays, and this explanation is at the present time adopted by
the majority of physicists ; but it is not necessary to consider the
phenomena under this aspect in order to establish the theory of heat.
In the course of this work it will be seen how the laws of equili
brium and propagation of radiant heat, in solid or liquid masses,

SECT. III.] PRINCIPLE OF COMMUNICATION. 41

can be rigorously demonstrated, independently of any physical
explanation, as the necessary consequences of common observations.

SECTION III.
Principle of the communication of heat

57. We now proceed to examine what experiments teach us
concerning the communication of heat.

If two equal molecules are formed of the same substance and
have the same temperature, each of them receives from the other
as much heat as it gives up to it ; their mutual action may then be
regarded as null, since the result of this action can bring about no
change in the state of the molecules. If, on the contrary, the first
is hotter than the second, it sends to it more heat than it receives
from it ; the result of the mutual action is the difference of these
two quantities of heat. In all cases we make abstraction of
the two equal quantities of heat which any two material points
reciprocally give up ; we conceive that the point most heated
acts only on the other, and that, in virtue of this action, the first
loses a certain quantity of heat which is acquired by the second.
Thus the action of two molecules, or the quantity of heat which
the hottest communicates to the other, is the difference of the two
quantities which they give up to each other.

58. Suppose that we place in air a solid homogeneous body,
whose different points have unequal actual temperatures ; each of
the molecules of which the body is composed will begin to receive
heat from those which are at extremely small distances, or will
communicate it to them. This action exerted during the same
instant between all points of the mass, will produce an infinitesi
mal resultant change in all the temperatures : the solid will ex
perience at each instant similar effects, so that the variations of
temperature will become more and more sensible.

Consider only the system of two molecules, m and n, equal and
extremely near, and let us ascertain what quantity of heat the
first can receive from the second during one instant : we may
then apply the same reasoning to all the other points which are

42 THEORY OF HEAT. [CHAP. I.

near enough to the point m, to act directly on it during the first
instant.

The quantity of heat communicated by the point n to the
point m depends on the duration of the instant, on the very small
distance between these points, on the actual temperature of each
point, and on the nature of the solid substance ; that is to say, if
one of these elements happened to vary, all the other remaining
the same, the quantity of heat transmitted would vary also. Now
experiments have disclosed, in this respect, a general result : it
consists in this, that all the other circumstances being the same,
the quantity of heat which one of the molecules receives from the
other is proportional to the difference of temperature of the two
molecules. Thus the quantity would be double, triple, quadruple, if
everything else remaining the same, the difference of the tempera
ture of the point n from that of the point m became double, triple,
or quadruple. To account for this result, we must consider that the
action of n on m is always just as much greater as there is a greater
difference between the temperatures of the two points : it is null,
if the temperatures are equal, but if the molecule n contains more
heat than the equal molecule m, that is to say, if the temperature
of in being v, that of n is v + A, a portion of the exceeding heat
will pass from n to m. Now, if the excess of heat were double, or,
which is the same thing, if the temperature of n were v + 2 A, the
exceeding heat would be composed of two equal parts correspond
ing to the two halves of the whole difference of temperature 2A ;
each of these parts would have its proper effect as if it alone
existed : thus the quantity of heat communicated by n to m would
be twice as great as when the difference of temperature is only A.
This simultaneous action of the different parts of the exceeding
heat is that which constitutes the principle of the communication
of heat. It follows from it that the sum of the partial actions, or
the total quantity of heat which m receives from n is proportional
to the difference of the two temperatures.

59. Denoting by v and v the temperatures of two equal mole
cules m and nt by pt their extremely small distance, and by dt, the
infinitely small duration of the instant, the quantity of heat which
m receives from n during this instant will be expressed by
(v —v)<f) (p) . dt. We denote by $ (p) a certain function of the

SECT. III.] PRINCIPLE OF COMMUNICATION. 43

distance p which, in solid bodies and in liquids, becomes nothing
when p has a sensible magnitude. The function is the same for
every point of the same given substance ; it varies with the nature
of the substance.

60. The quantity of heat which bodies lose through their sur
face is subject to the same principle. If we denote by a- the area,
finite or infinitely small, of the surface, all of whose points have
the temperature v, and if a represents the temperature of the
atmospheric air, the coefficient h being the measure of the ex
ternal conducibility, we shall have ah (v — a) dt as the expression
for the quantity of heat which this surface cr transmits to the air
during the instant dt.

When the two molecules, one of which transmits to the other
a certain quantity of heat, belong to the same solid, the exact
expression for the heat communicated is that which we have
given in the preceding article ; and since the molecules are
extremely near, the difference of the temperatures is extremely
small. It is not the same when heat passes from a solid body into
a gaseous medium. But the experiments teach us that if the
difference is a quantity sufficiently small, the heat transmitted is
sensibly proportional to that difference, and that the number h
may, in these first researches 1t be considered as having a constant
value, proper to each state of the surface, but independent of the
temperature.

61. These propositions relative to the quantity of heat com
municated have been derived from different observations. We
see first, as an evident consequence of the expressions in question,
that if we increased by a common quantity all the initial tempe
ratures of the solid mass, and that of the medium in which it is
placed, the successive changes of temperature would be exactly
the same as if this increase had not been made. Now this result
is sensibly in accordance with experiment ; it has been admitted
by the physicists who first have observed the effects of heat.

1 More exact la^vs of cooling investigated experimentally by Dulong and Petit
vrill be found in the Journal de VEcole Poll/technique, Tome xi. pp. 234—294,
Paris, 1820, or in Jamin, Cours de Physique, Le$on 47. [A. F.]

44 THEOHY OF HEAT. [CHAP. I.

62. If the medium is maintained at a constant temperature,
and if the heated body which is placed in that medium has
dimensions sufficiently small for the temperature, whilst falling
more and more, to remain sensibly the same at all points of the
body, it follows from the same propositions, that a quantity of heat
will escape at each instant through the surface of the body pro
portional to the excess of its actual temperature over that of the
medium. Whence it is easy to conclude, as will be seen in the
course of this work, that the line whose abscissae represent the
times elapsed, and whose ordinates represent the temperatures
corresponding to those times, is a logarithmic curve : now, ob
servations also furnish the same result, when the excess of the
temperature of the solid over that of the- medium is a sufficiently
small quantity.

63. Suppose the medium to be maintained at the constant
temperature 0, and that the initial temperatures of different
points a, b, c, d &c. of the same mass are a, ft, y, B &c., that at the
end of the first instant they have become a', ft', y, S' &c., that at
the end of the second instant they have become a", ft'', 7", 8" &c.,
and so on. We may easily conclude from the propositions enun
ciated, that if the initial temperatures of the same points had
been get, g/3, gy, g$ &c. (g being any number whatever), they
would have become, at the end of the first instant, by virtue of
the action of the different points, got.', gff, gy', g$ &c., and at the
end of the second instant, gen", g/3'-, gy", gS" &c., and so on. For
instance, let us compare the case when the initial temperatures
of the points, a, I, c, d &c. were a, ft, 7, B &c. with that in which
they are 2a, 2/5, 27, 2S &c., the medium preserving in both cases
the temperature 0. In the second hypothesis, the difference of
the temperatures of any two points whatever is double what it
was in the first, and the excess of the temperature of each point,
over that of each molecule of the medium, is also double ; con
sequently the quantity of heat which any molecule whatever
sends to any other, or that which it receives, is, in the second
hypothesis, double of that which it was in the first. The change
of temperature which each point suffers being proportional to the
quantity of heat acquired, it follows that, in the second case, this
change is double what it was in the first case. Now we have

SECT. IV.] UNIFORM LINEAR MOVEMENT. 4.5

supposed that the initial temperature of the first point, which was
a, became a' at the end of the first instant ; hence if this initial
temperature had been 2 a, and if all the other temperatures had
been doubled, it would have become 2 a'. The same would be the
case with all the other molecules b, c, d, and a similar result
would be derived, if the ratio instead of being 2, were any number
whatever g. It follows then, from the principle of the communica
tion of heat, that if we increase or diminish in any given ratio
all the initial temperatures, we increase or diminish in the same
ratio all the successive temperatures.

This, like the two preceding results, is confirmed by observa
tion. It could not have existed if the quantity of heat which
passes from one molecule to another had not been, actually, pro
portional to the difference of the temperatures.

64. Observations have been made with accurate instruments,
on the permanent temperatures at different points of a bar or of a
metallic ring, and on the propagation of heat in the same bodies
and in several other solids of the form of spheres or cubes. The
results of these experiments agree with those which are derived
from the preceding propositions. They would be entirely differ
ent if the quantity of heat transmitted from one solid molecule to
another, or to a molecule of air, were not proportional to the
excess of temperature. It is necessary first to know all the
rigorous consequences of this proposition; by it we determine the
chief part of the quantities which are the object of the problem.
By comparing then the calculated values with those given by
numerous and very exact experiments, we can easily measure the
variations of the coefficients, and perfect our first researches.

SECTION IV.
On the uniform and linear movement of heat.

Go. We shall consider, in the first place, the uniform move
ment of heat in the simplest case, which is that of an infinite
solid enclosed between two parallel planes.

We suppose a solid body formed of some homogeneous sub
stance to be enclosed between two parallel and infinite planes;

46 THEORY OF HEAT. [CHAP. I.

the lower plane A is maintained, by any cause whatever, at a
constant temperature a ; we may imagine for example that the
mass is prolonged, and that the plane A is a section common to
the solid and to the enclosed mass, and is heated at all its points
by a constant source of heat; the upper plane B is also main
tained by a similar cause at a fixed temperature b, whose value is
less than that of a ; the problem is to determine what would be
the result of this hypothesis if it were continued for an infinite
time,

If we suppose the initial temperature of all parts of this body
to be b, it is evident that the heat which leaves the source A will
be propagated farther and farther and will raise the temperature
of the molecules included between the two planes : but the tem
perature of the upper plane being unable, according to hypothesis
to rise above b} the heat will be dispersed within the cooler mass,
contact with which keeps the plane B at the constant temperature
b. The system of temperatures will tend more and more to a
final state, which it will never attain, but which would have the
property, as we shall proceed to shew, of existing and keeping
itself up without any change if it were once formed.

In the final and fixed state, which we are considering, the per
manent temperature of a point of the solid is evidently the same
at all points of the same section parallel to the base; and we
shall prove that this fixed temperature, common to all the points
of an intermediate section, decreases in arithmetic progression
from the base to the upper plane, that is to say, if we represent
the constant temperatures a and b by the ordinates AOL and Bj3

\

Fig. 1.

(see Fig. 1), raised perpendicularly to the distance AB between the
two planes, the fixed temperatures of the intermediate layers will
be represented by the ordinates of the straight line aft which

SECT. IV.] UNIFORM LINEAR MOVEMENT. 47

joins the extremities a. and /3; thus, denoting by z the height of
an intermediate section or its perpendicular distance from the
plane A, by e the whole height or distance AB, and by v the
temperature of the section whose height is z, we must have the

b — a
equation v = a -\ -- z.

6

In fact, if the temperatures were at first established in accord
ance with this law, and if the extreme surfaces A and B were
always kept at the temperatures a and b, no change would
happen in the state of the solid. To convince ourselves of this,
it will be sufficient to compare the quantity of heat which would
traverse an intermediate section A' with that which, during the
same time, would traverse another section B'.

Bearing in mind that the final state of the solid is formed
and continues, wre see that the part of the mass wrhich is below
the plane A' must communicate heat to the part which is above
that plane, since this second part is cooler than the first.

Imagine two points of the solid, m and m, very near to each
other, and placed in any manner whatever, the one m below the
plane A', and the other m above this plane, to be exerting their
action during an infinitely small instant : m the hottest point
will communicate to m a certain quantity of heat which will
cross the plane A'. Let x, y, z be the rectangular coordinates
of the point m, and x, y', z the coordinates of the point m :
consider also two other points n and n very near to each other,
and situated with respect to the plane B ', in the same manner
in which m and m are placed with respect to the plane A' : that
is to say, denoting by f the perpendicular distance of the two
sections A' and J57, the coordinates of the point n will be x, y, z + f
and those of the point n', x, y', zf + % ; the two distances mm
and nri will be equal : further, the difference of the temperature
v of the point m above the temperature v' of the point m will
be the same as the difference of temperature of the two points
n and n '. In fact the former difference will be determined by
substituting first z and then / in the general equation

b — a

and subtracting the second equation from the first, whence the

48 THEORY OF HEAT. [CHAP. I.

result v — v = "" a (z — z). We shall then find, by the sub-

Q

stitution of z + % and z + f, that the excess of temperature of
the point n over that of the point ri is also expressed by

Z> — a ,

It follows from this that the quantity of heat sent by the
point m to the point m will be the same as the quantity of heat
sent by the point n to the point ri, for all the elements which
concur in determining this quantity of transmitted heat are the
same.

It is manifest that we can apply the same reasoning to every
system of two molecules which communicate heat to each other
across the section A' or the section Bf ; whence, if we could
sum up the whole quantity of heat which flows, during the same
instant, across the section A' or the section J9', we should find
this quantity to be the same for both sections.

From this it follows that the part of the solid included be
tween Af and B' receives always as much heat as it loses, and
since this result is applicable to any portion whatever of the
mass included between two parallel sections, it is evident that
no part of the solid can acquire a temperature higher than that
which it has at present. Thus, it has been rigorously demon
strated that the state of the prism will continue to exist just as it
was at first.

Hence, the permanent temperatures of different sections of a
solid enclosed between two parallel infinite planes, are represented
by the ordinates of a straight line a/3, and satisfy the linear

b — a
equation v = a -\ --- z.

Q

66. By what precedes we see distinctly what constitutes
the propagation of heat in a solid enclosed between two parallel
and infinite planes, each of which is maintained at a constant
temperature. Heat penetrates the mass gradually across the
lower plane : the temperatures of the intermediate sections are
raised, but can never exceed nor even quite attain a certain
limit which they approach nearer and nearer : this limit or final
temperature is different for different intermediate layers, and

SECT. IV.]

UNIFORM LINEAR MOVEMENT.

decreases in arithmetic progression from the fixed temperature
of the lower plane to the fixed temperature of the upper plane.

The final temperatures are those which would have to be
given to the solid in order that its state might be permanent ;
the variable state which precedes it may also be submitted to
analysis, as we shall see presently: but we are now considering
only the system of final and permanent temperatures. In the
last state, during each division of time, across a section parallel
to the base, or a definite portion of that section, a certain
quantity of heat flows, which is constant if the divisions of time
are equal. This uniform flow is the same for all the intermediate
sections ; it is equal to that which proceeds from the source, and
to that which is lost during the same time, at the upper surface
of the solid, by virtue of the cause which keeps the temperature
constant.

67. The problem now is to measure that quantity of heat
which is propagated uniformly within the solid, during a given
time, across a definite part of a section parallel to the base : it
depends, as we shall see, on the two extreme temperatures a
and b, and on the distance e between the two sides of the solid ;
it would vary if any one of these elements began to change, the
other remaining the same. Suppose a second solid to be formed
of the same substance as the first, and enclosed between two

I'

Fig. 2.

infinite parallel planes, whose perpendicular distance is e (see
fig. 2) : the lower side is maintained at a fixed -temperature a ',
and the upper side at the fixed temperature &' ; both solids are
considered to be in that final and permanent state which has
the property of maintaining itself as soon as it has been formed.

F. H. 4

50 THEORY OF HEAT. [CHAP. I.

Thus the law of the temperatures is expressed for the first body

by the equation v = a H z, and for the second, by the equa
te

H — a

tion u = a H -, — z, v in the first solid, and u in the second, being

&

the temperature of the section whose height is z.

This arranged, we will compare the quantity of heat which,
during the unit of time traverses a unit of area taken on an
intermediate section L of the first solid, with that which during
the same time traverses an equal area taken on the section L'
of the second, e being the height common to the two sections,
that is to say, the distance of each of them from their own
base. We shall consider two very near points n and ri in the
first body, one of which n is below the plane L and the other
ri above this plane : x, y, z are the co-ordinates of n : and xf, y ', z
the co-ordinates of ri, e being less than z, and greater than z.

We shall consider also in the second solid the instantaneous
action of two points p and p, which are situated, with respect
to the section U, in the same manner as the points n and ri with
respect to the section L of the firsfc solid. Thus the same co
ordinates x, y, z, and of, y ', z referred to three rectangular axes
in the second body, will fix also the position of the points p
and p'.

Now, the distance from the point n to the point ri is equal
to the distance from the point p to the point p', and since the
two bodies are formed of the same substance, we conclude, ac
cording to the principle of the communication of heat, that the
action of n on ri, or the quantity of heat given by n to ri, and
the action of p on p ' , are to each other in the same ratio as the
differences of the temperature v — v and u — u.

Substituting v and then v' in the equation which belongs to

the first solid, and subtracting, we findv — v = (z — /) ; we

6

have also by means of the second equation u — u= — -, — (z — z'},

6

whence the ratio of the two actions in question is that of to

a'-V

e' '

SECT. IV.] UNIFORM LINEAR MOVEMENT.

51

We may now imagine many other systems of two molecules,
the first of which sends to the second across the plane L, a certain
quantity of heat, and each of these systems, chosen in the first
solid, may be compared with a homologous system situated in the
second, and whose action is exerted across the section L' ; we
can then apply again the previous reasoning to prove that the

a~b a ~~

ratio of the two actions is always that of - — - to

e e

Now, the whole quantity of heat which, during one instant,
crosses the section Z, results from the simultaneous action of a
multitude of systems each of which is formed of two points;
hence this quantity of heat and that which, in the second solid,
crosses during the same instant the section L', are also to each

other in the ratio of °^— to a ~ _
e e

It is easy then to compare with each other the intensities of
the constant flows of heat which are propagated uniformly in the
two solids, that is to say, the quantities of heat which, during
unit of time, "cross unit of surface of each of these bodies. The

ratio of these intensities is that of the two quotients a^~ and

a'-b'

~i — • If the two quotients are equal, the flows are the same,

whatever in other respects the values a, b} e, a, U, e, may be ;
in general, denoting the first flow by F and the second by F't

we shall have == = °^~— -r- a ~ .

68. Suppose that in the second solid, the permanent tempera
ture a of the lower plane is that of boiling water, 1 ; that the
temperature e of the upper plane is that of melting ice, 0; that
the distance e of the two planes is the unit of measure (a
metre); let us denote by K the constant flow of heat which,
during unit of time (a minute) would cross unit of surface in .
this last solid, if it were formed of a given substance ; K ex- (
pressing a certain number of units of heat, that~is to say a certain
number of times the heat necessary to convert a kilogramme
of ice into water : we shall have, in general, to determine the

4—2

52 THEORY OF HEAT. [CHAP. I.

constant flow F, in a solid formed of the same substance, the

F a-b w a-b
equation ^— - - or H — A .

J\. & 6

The value of F denotes the quantity of heat which, during
the unit of time, passes across a unit of area of the surface taken
on a section parallel to the base.

Thus the thermometric state of a solid enclosed between two
parallel infinite plane sides whose perpendicular distance is e,
and which are maintained at fixed temperatures a and b, is
represented by the two equations :

b — a a-b ^ T^dv

v = a + zt and F=K- — - or F=-K-^.

The first of these equations expresses the law according to
which the temperatures decrease from the lower side to the
opposite side, the second indicates the quantity of heat which,
during a given time, crosses a definite part of a section parallel
to the base.

69. We have taken this coefficient K, which enters into
the second equation, to be the measure of the specific conduci
bility of each substance ; this number has very different values
for different bodies.

It represents, in general, the quantity of heat which, in a
homogeneous solid formed of a given substance and enclosed
between two infinite parallel planes, flows, during one minute,
across a surface of one square metre taken on a section parallel
to the extreme planes, supposing that these two planes are main
tained, one at the temperature of boiling water, the other at
the temperature of melting ice, and that all the intermediate
planes have acquired and retain a permanent temperature.

We might employ another definition of conducibility, since
we could estimate the capacity for heat by referring it to unit
of volume, instead of referring it to unit of mass. All these
definitions are equally good provided they are clear and pre
cise.

We shall shew presently how to determine by observation the
value K of the conducibility or conductibility in different sub
stances.

SECT. IV.] UNIFORM LINEAR MOVEMENT. 53

70. In order to establish the equations which we have
cited in Article 68, it would not -be necessary to suppose the
points which exert their action across the planes to be at ex
tremely small distances.

^ The results would still be the same if the distances of these
points had any magnitude whatever ; they would therefore apply
also to the case where the direct action of heat extended within
the interior of the mass to very considerable distances, all the
circumstances which constitute the hypothesis remaining in other
respects the same.

We need only suppose that the cause which maintains the
temperatures at the surface of the solid, affects not only that
part of the mass which is extremely near to the surface, but that

its action extends to a finite depth. The equation V = a-a~b2

e

will still represent in this case the permanent temperatures of
the solid. The true sense of this proposition is that, if we give
to all points of the mass the temperatures expressed by the
equation, and if besides any cause whatever, acting on the two
extreme laminae, retained always every one of their molecules
at the temperature which the same equation assigns to them,
the interior points of the solid would preserve without any change
their initial state.

If we supposed that the action of a point of the mass could
extend to a finite distance e, it would be necessary that the
thickness of the extreme laminae, whose state is maintained by
the external cause, should be at least equal to e. But the
quantity e having in fact, in the natural state of solids, only
an inappreciable value, we may make abstraction of this thick
ness; and it is sufficient for the external cause to act on each
of the two layers, extremely thin, which bound the solid. This
is always what must be understood by the expression, to maintain
the temperature of the surface constant.

71. We proceed further to examine the case in which the
same solid would be exposed, at one of its faces, to atmospheric
air maintained at a constant temperature.

Suppose then that the lower plane preserves the fixed tem
perature a, by virtue of any external cause whatever, and that

54 THEORY OF HEAT. [CHAP. I.

the upper plane, instead of being maintained as formerly at a
less temperature b, is exposed to atmospheric air maintained
at that temperature b, the perpendicular distance of the two
planes being denoted always by e : the problem is to determine
the final temperatures.

Assuming that in the initial state of the solid, the common
temperature of its molecules is b or less than b, we can readily
imagine that the heat which proceeds incessantly from the source
A penetrates the mass, and raises more and more the tempera
tures of the intermediate sections ; the upper surface is gradually
heated, and permits part of the heat which has penetrated the
solid to escape into the air. The system of temperatures con
tinually approaches a final state which would exist of itself if
it were once formed; in this final state, which is that which
we are considering, the temperature of the plane B has a fixed
but unknown value, which we will denote by ft, and since the
lower plane A preserves also a permanent temperature a, the
system of temperatures is represented by the general equation

v = a + - z, v denoting always the fixed temperature of the

section whose height is z. The quantity of heat which flows
during unit of time across a unit of surface taken on any section

whatever is fr - , % denoting the interior conducibility.

We must now consider that the upper surface B, whose
temperature is ft, permits the escape into the air of a certain
quantity of heat which must be exactly equal to that which
crosses any section whatever L of the solid. If it were not so,
the part of the mass included between this section L and the
plane B would not receive a quantity of heat equal to that
which it loses; hence it would not maintain its state, which is
contrary to hypothesis ; the constant flow at the surface is there
fore equal to that which traverses the solid : now, the quantity
of heat which escapes, during unit of time, from unit of surface
taken on the plane B, is expressed by li(ft-b), b being the
fixed temperature of the air, and h the measure of the conduci
bility of the surface B\ we must therefore have the equation

V~T~ = h(@- b), which will determine the value of ft.

SECT. IV.] UNIFORM LINEAR MOVEMENT. 55

From this may be derived a — /3= — p j-~ » an equation

fl6 ~\~ K

whose second member is known ; for the temperatures a and 6
are given, as are also the quantities h, ^, e.

Introducing this value of a- ft into the general equation

v = a + - —z, we shall have, to express the temperatures of any
section of the solid, the equation a — v=-^~ j - , in which

llG ~r~ rC

known quantities only enter with the corresponding variables v
and z.

72. So far we have determined the final and permanent state
of the temperatures in a solid enclosed between two infinite and
parallel plane surfaces, maintained at unequal temperatures.
This first case is, properly speaking, the case of the linear and
uniform propagation of heat, for there is no transfer of heat in
the plane parallel to the sides of the solid ; that which traverses
the solid flaws uniformly, since the value of the flow is the same
for all instants and for all sections.

We will now restate the three chief propositions which result
from the examination of this problem ; they are susceptible of a
great number of applications, and form the first elements of our
theory.

1st. If at the two extremities of the thickness e of the solid
we erect perpendiculars to represent the temperatures a and b
of the two sides, and if we draw the straight line which joins
the extremities of these two first ordinates, all the intermediate
temperatures will be proportional to the ordinates of this straight

line ; they are expressed by the general equation a — v = - - z,

6

v denoting the temperature of the section whose height is z.

2nd. The quantity of heat which flows uniformly, during
unit of time, across unit of surface taken on any section whatever
parallel to the sides, all other things being equal, is directly
proportional to the difference a — b of the extreme temperatures,
and inversely proportional to the distance e which separates

^a-6
these sides. The quantity of heat is expressed by K - — , or

56 THEORY OF HEAT. [CHAP. I.

— K — , if we derive from the general equation the value of

-v- which is constant; this uniform flow may always be repre
sented, for a given substance and in the solid under examination,
by the tangent of the angle included between the perpendicular
e and the straight line whose ordinates represent the tempera
tures.

3rd. One of the extreme surfaces of the solid being submitted
always to the temperature a, if the other plane is exposed to air
maintained at a fixed temperature b ; the plane in contact with
the air acquires, as in the preceding case, a fixed temperature /?,
greater than b, and it permits a quantity of heat to escape into
the air across unit of surface, during unit of time, which is ex
pressed by h (/3 — b) , h denoting the external conducibility of
the plane.

The same flow of heat h(/3 — b) is equal to that which
traverses the prism and whose value is K(a — ft)\ we have there
fore the equation h({3 — ft) = K — — , which gives the value
of

SECTION V.

Law of the permanent temperatures in a prism of small

thickness.

73. We shall easily apply the principles which have just
been explained to the following problem, very simple in itself,
but one whose solution it is important to base on exact theory.

A metal bar, whose form is that of a rectangular parallele
piped infinite in length, is exposed to the action of a source of
heat which produces a constant temperature at all points of its
extremity A. It is required to determine the fixed temperatures
at the different sections of the bar.

The section perpendicular to the axis is supposed to be a
square whose side 21 is so small that we may without sensible
error consider the temperatures to be equal at different points
of the same section. The air in which the bar is placed is main-

SECT. V.] STEADY TEMPERATURE IN A BAR. 57

tained at a constant temperature 0, and carried away by a
current with uniform velocity.

Within the interior of the solid, heat will pass successively
all the parts situate to the right^of the source, and not exposed
directly to its action; they will be heated more and more, but
the temperature of each point will not increase beyond a certain
limit. This maximum temperature is not the same for every
section ; it in general decreases as the distance of the section
from the origin increases : we shall denote by v the fixed tem
perature of a section perpendicular to the axis, and situate at a
distance x from the origin A

Before every point of the solid has attained its highest degree
of heat, the system of temperatures varies continually, and ap
proaches more and more to a fixed state, which is that which
we consider. This final state is kept up of itself when it has
once been formed. In order that the system of temperatures
may be permanent, it is necessary that the quantity of heat
which, during unit of time, crosses a section made at a distance x
from the origin, should balance exactly all the heat which, during
the same time, escapes through that part of the external surface
of the prism which is situated to the right of the same section.
The lamina whose thickness is dx, and whose external surface
is Sldx, allows the escape into the air, during unit of time, of
a quantity of beat expressed by Shlv . dx, h being the measure of
the external conducibility of the prism. Hence taking the in
tegral jShlv . dx from x = 0 to x — oo , we shall find the quantity
of heat wrhich escapes from the whole surface of the bar durino-
unit of time ; and if we take the same integral from x = 0 to
x = x, we shall have the quantity of heat lost through the part
of the surface included between the source of heat and the section
made at the distance x. Denoting the first integral by (7, whose
value is constant, and the variable value of the second by
jShlv.dx-, the difference C-/8hlv.dx will express the whole
quantity of heat which escapes into the air across the part of
the surface situate to the right of the section. On the other
hand, the lamina of the solid, enclosed between two sections
infinitely near at distances x and x + dx, must resemble an in
finite solid, bounded by two parallel planes, subject to fixed
temperatures v and v + dv, since, by hypothesis, the temperature

58 THEORY OF HEAT. [CHAP. I.

does not vary throughout the whole extent of the same section.
The thickness of the solid is dx, and the area of the section is
4/2 : hence the quantity of heat which flows uniformly, during
unit of time, across a section of this solid, is, according to the

preceding principles, — 4Z2A -=- , k being the specific internal con-
ducibility : we must therefore have the equation

V"

whence

^ \\\ i

74. We should obtain the same result by considering the
equilibrium of heat in a single lamina infinitely thin, enclosed
between two sections at distances x arid x + dx. In fact, the
quantity of heat which, during unit of time, crosses the first

section situate at distance x, is — 4/2X -r- . To find that which

flows during the same time across the successive section situate
at distance x + dx, we must in the preceding expression change x

into x + dx, which gives — 4Z2&. ^~ + d ~ . If we subtract

[dx \dxjj

the second expression from the first we shall find how much
heat is acquired by the lamina bounded by these two sections
during unit of time ; and since the state of the lamina is per
manent, it follows that all the heat acquired is dispersed into
the air across the external surface Sldx of the same lamina : now
the last quantity of heat is Shlvdx : we shall obtain therefore the
same equation

07 7 7 ^727 7 A&A 1 ^V 27?,

8/uvdx — klkd -y- , whence -^—5 = -=-=• v.
\dxj ' dx2 kl

75. In whatever manner this equation is formed, it is
necessary to remark that the quantity of heat which passes into
the lamina whose thickness is dx, has a finite value, and that

its exact expression is — 4<l2k ^- . The lamina being enclosed
between two surfaces the first of which has a temperature v,

SECT. V.] STEADY TEMPERATURE IX A BAR. 59

and the second a lower temperature v', we see that the quantity
of heat which it receives through the first surface depends on
the difference v — v', and is proportional to it : but this remark
is not sufficient to complete the calculation. The quantity in
question is not a differential : it has a finite value, since it is
equivalent to all the heat which escapes through that part of
the external surface of the prism which is situate to the right
of the section. To form an exact idea of it, we must compare
the lamina whose thickness is dx, with a solid terminated by
two parallel planes whose distance is e, and which are maintained
at unequal temperatures a and b. The quantity of heat which
passes into such a prism across the hottest surface, is in fact
proportional to the difference a — b of the extreme temperatures,
but it does not depend only on this difference : all other things
being equal, it is less when the prism is thicker, and in general

it is proportional to . This is why the quantity of heat

^

which passes through the first surface into the lamina, whose

thickness is dx} is proportional to —-= — .

dx

We lay stress on this remark because the neglect of it has
been the first obstacle to the establishment of the theory. If
we did not make a complete analysis of the elements of the
problem, we should obtain an equation not homogeneous, and,
a fortiori, we should not be able to form the equations which
express the movement of heat in more complex cases.

It was necessary also to introduce into the calculation the
dimensions of the prism, in order that we might not regard, as
general, consequences which observation had furnished in a par
ticular case. Thus, it was discovered by experiment that a bar
of iron, heated at one extremity, could not acquire, at a distance
of six feet from the source, a temperature of one degree (octo-
gesimal1) ; for to produce this effect, it would be necessary for
the heat of the source to surpass considerably the point of fusion
of iron; but this result depends on the thickness of the prism*
employed. If it had been greater, the heat would have been,
propagated to a greater distance, that is to say, the point of
the bar which acquires a fixed temperature of one degree is
1 Reaumur's Scale of Temperature. [A. F.J

60 THEORY OF HEAT. [CHAP. I.

much more remote from the source when the bar is thicker, all
other conditions remaining the same. We can always raise by
one degree the temperature of one end of a bar of iron, by heating
the solid at the other end ; we need only give the radius of the
base a sufficient length : which is, we may say, evident, and
of which besides a proof will be found in the solution of the
problem (Art. 78).

76. The integral of the preceding equation is

A and B being two arbitrary constants ; now, if we suppose the
distance x infinite, the value of the temperature v must be

75

+x*

infinitely small; hence the term Be+x* w does not exist in the in-

/2k

tegral : thus the equation v = Ae~*'^ u represents the permanent
state of the solid ; the temperature at the origin is denoted by
the constant At since that is the value of v when x is zero.

This law according to which the temperatures decrease
is the same as that given by experiment ; several physicists
have observed the fixed temperatures at different points of a
metal bar exposed at its extremity to the constant action of a
source of heat, and they have ascertained that the distances
from the origin represent logarithms, and the temperatures the
corresponding numbers.

77. The numerical value of the constant quotient of two con
secutive temperatures being determined by observation, we easily

deduce the value of the ratio -; for, denoting by vlt va the tem
peratures corresponding to the distances x^ x2, we have

v ~{*i-*tk/s -i /2h log v — loof v9 ,,

~* = e v **, whence A / --=- = & 1 — * Jl.

v A/ k x x

As for the separate values of li and k, they cannot be deter
mined by experiments of this kind : we must observe also the
varying motion of heat.

78. Suppose two bars of the same material and different
dimensions to be submitted at their extremities to the same tern-

SECT. V.] STEADY TEMPERATURE IX A BAR. 61

perature A ; let lt be the side of a section in the first bar, and 12
iii the second, we shall have, to express the temperatures of these
two solids, the equations

Vl = Ae~1 and v9=Ae~

i\, in the first solid, denoting the temperature of a section made
at distance xlf and vz, in the second solid, the temperature of a
section made at distance xz.

When these two bars have arrived at a fixed state, the tem
perature of a section of the first, at a certain distance from the
source, will not be equal to the temperature of a section of the
second at the same distance from the focus ; in order that the
fixed temperatures may be equal, the distances must be different.
If we wish to compare with each other the distances x^ and x<2
from the origin up to the points which in the two bars attain
the same temperature, we must equate the second members of

these equations, and from them we conclude that -\ = j. Thus

xz '2

the distances in question are to each other as the square roots of
the thicknesses.

79. If two metal bars of equal dimensions, but formed of
different substances, are covered with the same coating, which
gives them the same external conducibility1, and if they are
submitted at their extremities to the same temperature, heat will
be propagated most easily and to the greatest distance from the
origin in that which has the greatest conducibility. To compare
with each other the distances xl and xz from the common origin
up to the points which acquire the same fixed temperature, we
must, after denoting the respective conducibilities of the two
substances by k^ and &2, write the equation

/** /^ -r2 I*

e-W^ = e-W w f whence ^ = p .

x* k2

Thus the ratio of the two conducibilities is that of the squares
of the distances from the common origin to the points which
attain the same fixed temperature.

1 Ingenhousz (1789), Sur les mgtaux comme conducteurs de la chalenr. Journal
de Physique, xxxiv., 68, 380. Gren's Journal der Physik, Bd. I." [A. F.]

C2 THEORY OF HEAT. [CHAP. I.

80. It is easy to ascertain how much heat flows during unit
of time through a section of the bar arrived at its fixed state :

7 I2A

this quantity is expressed by — 4K2-y- , or kAj±khl*.e *Kj and

if we take its value at the origin, we shall have bAjZkh? as the
measure of the quantity of heat which passes from the source
into the solid during unit of time ; thus the expenditure of the
source of heat is, all other things being equal, proportional to the
square root of the cube of the thickness.

We should obtain the same result on taking the integral
fShlv . dx from x nothing to x infinite.

SECTION VI.
On the heating of closed spaces.

81. We shall again make use of the theorems of Article 72
in the following problem, whose solution offers useful applications ;
it consists in determining the extent of the heating of closed
spaces.

Imagine a closed space, of any form whatever, to be filled with
atmospheric air and closed on all sides, and that all parts of the
boundary are homogeneous and have a common thickness e, so
small that the ratio of the external surface to the internal surface
differs little from unity. The space which this boundary termi
nates is heated by a source whose action is constant ; for example,
by means of a surface whose area is cr maintained at a constant
temperature a.

We consider here only the mean temperature of the air con
tained in the space, without regard to the unequal distribution of
heat in this mass of air ; thus we suppose that the existing causes
incessantly mingle all the portions of air, and make their tem
peratures uniform.

We see first that the heat which continually leaves the source
spreads itself in the surrounding air and penetrates the mass of
which the boundary is formed, is partly dispersed at the surface,

SECT. VJ.] HEATING OF CLOSED SPACES. 63

and passes into the external air, which we suppose to be main
tained at a lower and permanent temperature n. The inner air is
heated more and more : the same is the case with the solid
boundary : the system of temperatures steadily approaches a final
state which is the object of the problem, and has the property of
existing by itself and of being kept up unchanged, provided the
surface of the source a be maintained at the temperature a, and
the external air at the temperature n.

In the permanent state which we wish to determine the air
preserves a fixed temperature m ; the temperature of the inner
surface s of the solid boundary has also a fixed value a ; lastly, the
outer surface s, which terminates the enclosure, preserves a fixed
temperature b less than a, but greater than n. The quantities
cr, a, 5, e and n are known, and the quantities m, a and b are
unknown.

The degree of heating consists in the excess of the temperature
m over n} the temperature of the external air; this excess evi
dently depends on the area a of the heating surface and on its
temperature a ; it depends also on the thickness e of the en
closure, on the area s of the surface which bounds it, on the
facility with which heat penetrates the inner surface or that
which is opposite to it ; finally, on the specific conducibility of
the solid mass which forms the enclosure : for if any one of these
elements were to be changed, the others remaining the same, the
degree of the heating would vary also. The problem is to deter
mine how all these quantities enter into the value of m — n.

82. The solid boundary is terminated by two equal surfaces,
each of which is maintained at a fixed temperature; every
prismatic element of the solid enclosed between two opposite por
tions of these surfaces, and the normals raised round the contour
of the bases, is therefore in the same state as if it belonged to an
infinite solid enclosed between two parallel planes, maintained at
unequal temperatures. All the prismatic elements which com
pose the boundary touch along their whole length. The points
of the mass which are equidistant from the inner surface have
equal temperatures, to whatever prism they belong ; consequently
there cannot be any transfer of heat in the direction perpendicular
to the length of these prisms. The case is, therefore, the same

64 THEORY OF HEAT. [CHAP. I.

as that of which we have already treated, and we must apply
to it the linear equations which have been stated in former
articles.

83. Thus in the permanent state which we are considering,
the flow of heat which leaves the surface cr during a unit of time,
is equal to that which, during the same time, passes from the
surrounding air into the inner surface of the enclosure ; it is
equal also to that which, in a unit of time, crosses an inter
mediate section made within the solid enclosure by a surface
equal and parallel to those which bound this enclosure ; lastly,
the same flow is again equal to that which passes from the solid
enclosure across its external surface, and is dispersed into the air.
If these four quantities of flow of heat were not equal, some
variation would necessarily occur in the state of the temperatures,
which is contrary to the hypothesis.

The first quantity is expressed by a (a. — m) g, denoting by
g the external conducibility of the surface cr, which belongs to
the source of heat.

The second is s (m — a) h, the coefficient h being the measure
of the external conducibility of the surface s, which is exposed
to the action of the source of heat.

The third is s — — K, the coefficient K being the measure of

6

the conducibility proper to the homogeneous substance which
forms the boundary.

The fourth is s(b — n}H, denoting by H the external con
ducibility of the surface s, which the heat quits to be dispersed
into the air. The coefficients h and H may have very unequal
values on account of the difference of the state of the two surfaces
which bound the enclosure ; they are supposed to be known, as
also the coefficient K: we shall have then, to determine the three
unknown quantities m, a and 6, the three equations :

f N a — b r,
a (a — m) g = s - A,

G

cr (a - m) g = s (b — n) H.

SECT. VI.] HEATING OF CLOSED SPACES. 65

84. The value of m is the special object of the problem. It
may be found by writing the equations in the form

adding, we have m — n = (a. - m) P,

denoting by P the known quantity ^ (| -f ^ -f J^ J ;

whence we conclude

m — 11 = a — n

85. The result shews how m — n, the extent of the heating,
depends on given quantities which constitute the hypothesis.
We will indicate the chief results to be derived from it \

1st. The extent of the heating m — n is directly proportional
to the excess of the temperature of the source over that of the
external air.

2nd: The value of m — n does not depend on the form of
the enclosure nor on its volume, but only on the ratio - of the

surface from which the heat proceeds to the surface which receives
it, and also on e the thickness of the boundary.

If we double cr the surface of the source of heat, the extent
of the heating does not become double, but increases according
to a certain law which the equation expresses.

1 These results \vere stated by the author in a rather different manner in the
extract from his original memoir published in the Bulletin par la Society Philo-
matique de Paris, 1818, pp. 1—11. [A. F.]

F. H. 5

66 THEORY OF HEAT. [CHAP. I.

3rd. All the specific coefficients which regulate the action
of the heat, that is to say, g, K, H and h, compose, with the

dimension e, in the value of m — n a single element f + 77+ fr>

whose value may be determined by observation.

If we doubled e the thickness of the boundary, we should
have the same result a>s if, in forming it, we employed a sub
stance whose conducibility proper was twice as great. Thus the
employment of substances which are bad conductors of heat
permits us to make the thickness of the boundary small; the

o

effect which is obtained depends only on the ratio - .

4th. If the conducibility K is nothing, we find
that is to say, the inner air assumes the temperature of the
source : the same is the case if H is zero, or h zero. These con
sequences are otherwise evident, since the heat cannot then be
dispersed into the external air.

5th. The values of the quantities g, H, h, K and a, which
we supposed known, may be measured by direct experiments,
as we shall shew in 'the sequel ; but in the actual problem, it
will be sufficient to notice the value of m — n which corresponds
to given values of cr and of a, and this value may be used to

determine the whole coefficient j- + ^ + jj. , by means of the equa-

ii/ j\. jj.

tion m — n — (a — n}-p~ (1 +- p] in which p denotes the co
efficient sought. We must substitute in this equation, instead

of - and a — n, the values of those quantities, which we suppose
s

given, and that of m — n which observation will have made
known. From it may be derived the value of p, and we may
then apply the formula to any number of other cases.

6th. The coefficient H enters into the value of m — n in
the same manner as the coefficient h; consequently the state of
the surface, or that of the envelope which covers it, produces
the same effect, whether it has reference to the inner or outer
surface.

We should have considered it useless to take notice of these

SECT. VI.] HEATING OF CLOSED SPACES. 67

different consequences, if we were not treating here of entirely
new problems, whose results may be of direct use.

86. We know that animated bodies retain a temperature
sensibly fixed, which we may regard as independent of the tem
perature of the medium in which they live. These bodies are,
after some fashion, constant sources of heat, just as inflamed
substances are in which the combustion has become uniform.
We may then, by aid of the preceding remarks, foresee and
regulate exactly the rise of temperature in places where a great
number of men are collected together. If we there observe the
height of the thermometer under given circumstances, we shall
determine in advance what that height would be, if the number
of men assembled in the same space became very much greater.

In reality, there are several accessory circumstances which
modify the results, such as the unequal thickness of the parts
of the enclosure, the difference of their aspect, the effects which
the outlets produce, the unequal distribution of heat in the air.
We cannot therefore rigorously apply the rules given by analysis ;
nevertheless these rules are valuable in themselves, because they
contain the tine principles of the matter : they prevent vague
reasonings and useless or confused attempts.

87. If the same space were heated by two or more sources
of different kinds, or if the first inclosure were itself contained
in a second enclosure separated from the first by a mass of air,
we might easily determine in like manner the degree of heating
and the temperature of the surfaces.

If we suppose that, besides the first source u, there is a second
heated surface TT, whose constant temperature is y&, and external
conducibility j, we shall find, all the other denominations being
retained, the following equation :

\

m — n= -

n^jfe t I t l\
K+H + h)

_
s \& H h

If we suppose only one source a; and if the first enclosure is
itself contained in a second, s, h', K' , H', e, representing the

5—2

68 THEORY OF HEAT. [CHAP. I.

elements of the second enclosure which correspond to those of
the first which were denoted by 5, h, K, H, e ; we shall find,
p denoting the temperature of the air which surrounds the ex
ternal surface of the second enclosure, the following equation :

The quantity P represents

* (9 ±9* +

7 r + j^^

s \li K.

We should obtain a similar result if we had three or a greater
number of successive enclosures ; and from this we conclude that
these solid envelopes, separated by air, assist very much in in
creasing the degree of heating, however small their thickness
may be.

88. To make this remark more evident, we will compare the
quantity of heat which escapes from the heated ^surface, with
that which the same body would lose, if the surface which en
velopes it were separated from it by an interval filled with air.

If the body A be heated by a constant cause, so that its
surface preserves a fixed temperature b, the air being maintained
at a less temperature a, the quantity of heat which escapes into
the air in the unit of time across a unit of surface will be
expressed by h (b — a), h being the measure of the external con-
ducibility. Hence in order that the mass may preserve a fixed
temperature b, it is necessary that the source, whatever it may
be, should furnish a quantity of heat equal to hS (b — a), S de
noting the area of the surface of the solid.

Suppose an extremely thin shell to be detached from the
body A and separated from the solid by an interval filled with
air; and suppose the surface of the same solid A to be still
maintained at the temperature b. We see that the air contained
between the shell and the body will be heated and will take
a temperature a greater than a. The shell itself will attain
a permanent state and will transmit to the external air whose
fixed temperature is a all the heat which the body loses. It
follows that the quantity of heat escaping from the solid will

SECT. VI.] HEATING OF CLOSED SPACES. 69

be hS(b — aJ'}, instead of being hS(b — a), for we suppose that
the new surface of the solid and the surfaces which bound the
shell have likewise the same external conducibility h. It is
evident that the expenditure of the source of heat will be less
than it was at first. The problem is to determine the exact ratio
of these quantities.

89. Let e be the thickness of the shell, m the fixed tempera
ture of its inner surface, n that of its outer surface, and K its
internal conducibility. We shall have, as the expression of the
quantity of heat which leaves the solid through its surface,
hS(b-a').

As that of the quantity which penetrates the inner surface
of the shell, hS (a - m).

As that of the quantity which crosses any section whatever

of the same shell. KS .

e

Lastly, as the expression of the quantity which passes through
the outer surface into the air, hS (n — a).

All these quantities must be equal, we have therefore the
following equations :

rr

h (n — a) = —(m — ri),

h(n — a) = h (a — m),
h(n-a)=h(b-a).
If moreover we write down the identical equation

k(n — a) = h(n — a),
and arrange them all under the forms
n — a = n — a,

m-n = - (n-a)

I

b — a = n — a,
we find, on addition,

70 THEORY OF HEAT. [CHAP. I.

The quantity of heat lost by the solid was hS(b — a), when
its surface communicated freely with the air, it is now hS (6 — a)

or hS(n — a), which is equivalent to hS

The first quantity is greater than the second in the ratio of

In order therefore to maintain at temperature b a solid whose
surface communicates directly to the air, more than three times
as much heat is necessary than would be required to maintain
it at temperature Z>, when its extreme surface is not adherent
but separated from the solid by any small interval whatever filled
with air.

If we suppose the thickness e to be infinitely small, the
ratio of the quantities of heat lost will be 3, which would also
be the value if K were infinitely great.

We can easily account for this result, for the heat being
unable to escape into the external air, without penetrating several
surfaces, the quantity which flows out must diminish as the
number of interposed surfaces increases ; but we should have
been unable to arrive at any exact judgment in this case, if the
problem had not been submitted to analysis.

90. We have not considered, in the preceding article, the
effect of radiation across the layer of air which separates the
two surfaces ; nevertheless this circumstance modifies the prob
lem, since there is a portion of heat which passes directly across
the intervening air. We shall suppose then, to make the object
of the analysis more distinct, that the interval between the sur
faces is free from air, and that the heated body is covered by
any number whatever of parallel laminse separated from each
other.

If the heat which escapes from the solid through its plane
superficies maintained at a temperature b expanded itself freely
in vacuo and was received by a parallel surface maintained at
a less temperature a, the quantity which would be dispersed in
unit of time across unit of surface would be proportional to (b — a),
the difference of the two constant temperatures : this quantity

SECT. VI.] HEATING OF CLOSED SPACES. 71

would be represented by H (b — a), H being the value of the rela
tive conducibility which is not the same as h.

The source which maintains the solid in its original state must
therefore furnish, in every unit of time, a quantity of heat equal
toHS(b-a).

We must now determine the new value of this expenditure
in the case where the surface of the body is covered by several
successive laminae separated by intervals free from air, supposing
always that the solid is subject to the action of any external
cause whatever which .maintains its surface at the temperature b.

Imagine the whole system of temperatures to have become
fixed ; let m be the temperature of the under surface of the first
lamina which is consequently opposite to that of the solid, let n
be the temperature of the upper surface of the same lamina,
e its thickness, and K its specific conducibility ; denote also by
77&J, nlt m2, n2, m3, ??3, ??i4, w4, &c. the temperatures of the under
and upper surfaces of the different laminae, and by K} e, the con
ducibility and thickness of the same laminae; lastly, suppose all
these surfaces to be in a state similar to the surface of the solid,
so that the value of the coefficient H is common to them.

The quantity of heat which penetrates the under surface of
a lamina corresponding to any suffix i is HSfyi^—mJ), that which

J7-Q

crosses this lamina is — (mi~ ni)f an(^ the quantity which escapes
c

from its upper surface is HS(nt — mi+l}. These three quantities,
and all those which refer to the other laminae are equal ; we may
therefore form the equation by comparing all these quantities
in question with the first of them, which is HS (b — mj ; we shall
thus have, denoting the number of laminae \>y j :

He n

i - ni = ^ (b ~

He ,, .
- n, = (b - IflJ,

72 THEOKY OF HEAT. [CHAP. I.

He n
m*-n*=~K ^~m^

rij— a = b — m1.
Adding these equations, we find

The expenditure of the source of heat necessary to maintain
the surface of the body A at the temperature b is US (b — a),
when this surface sends its rays to a fixed surface maintained at
the temperature a. The expenditure is HS (b — m^ when we place
between the surface of the body A, and the fixed surface maintained
at temperature a, a numberj of isolated laminae; thus the quantity
of heat which the source must furnish is very much less in the
second hypotheses than in the first, and the ratio of the two

quantities is „ . If we suppose the thickness e of the

laminae to be infinitely small, the ratio is -. The expenditure

f+i
of the source is then inversely as the number of laminae which

cover the surface of the solid.

91. The examination of these results and of those which we
obtained when the intervals between successive enclosures were
occupied by atmospheric air explain clearly why the separation
of surfaces and the intervention of air assist very much in re
taining heat.

Analysis furnishes in addition analogous consequences when
we suppose the source to be external, and that the heat which
emanates from it crosses successively different diathermanous
envelopes and the air which they enclose. This is what has
happened when experimenters have exposed to the rays of the
sun thermometers covered by several sheets of glass within which
different layers of air have been enclosed.

For similar reasons the temperature of the higher regions
of the atmosphere is very much less than at the surface of the
earth.

SECT. VII.] MOVEMENT IX THREE DIMENSIONS. 73

In general the theorems concerning the heating of air in
closed spaces extend to a great variety of problems. It would
be useful to revert to them when we wish to foresee and regulate
temperature with precision, as in the case of green-houses, drying-
houses, sheep-folds, work-shops, or in many civil establishments,
such as hospitals, barracks, places of assembly.

In these different applications we must attend to accessory
circumstances which modify the results of analysis, such as the
unequal thickness of different parts of the enclosure, the intro
duction of air, &c. ; but these details would draw us away from
our chief object, which is the exact demonstration of general
principles.

For the rest, we have considered only, in what has just been
said, the permanent state of temperature in closed spaces. AVe
can in addition express analytically the variable state which
precedes, or that which begins to take place when the source of
heat is withdrawn, and we can also ascertain in this way, how
the specific properties of the bodies which we employ, or their
dimensions affect the progress and duration of the heating ; but
these researches require a different analysis, the principles of
which will be explained in the following chapters.

SECTION VII.

On the uniform movement of heat in three dimensions.

92. Up to this time we have considered the uniform move
ment of heat in one dimension only, but it is easy to apply the
same principles to the case in which heat is propagated uniformly
in three directions at right angles.

Suppose the different points of a solid enclosed by six planes
at right angles to have unequal actual temperatures represented
by the linear equation v = A -f ax + by + cz, x, y, z, being the
rectangular co-ordinates of a molecule whose temperature is v.
Suppose further that any external causes whatever acting on the
six faces of the prism maintain every one of the molecules situated
on the surface, at its actual temperature expressed by the general

equation

v — A -f ax + by + cz (a),

74 THEORY OF HEAT. [CHAP. I.

we shall prove that the same causes which, by hypothesis, keep
the outer layers of the solid in their initial state, are sufficient
to preserve also the actual temperatures of every one of the inner
molecules, so that their temperatures do not cease to be repre
sented by the linear equation.

The examination of this question is an element of the
general theory, it will serve to determine the laws of the varied
movement of heat in the interior of a solid of any form whatever,
for every one of the prismatic molecules of which the body is
composed is during an infinitely small time in a state similar
to that which the linear equation (a) expresses. We may then,
by following the ordinary principles of the differential calculus,
easily deduce from the notion of uniform movement the general
equations of varied movement.

93. In order to prove that when the extreme layers of the
solid preserve their temperatures no change can happen in the
interior of the mass, it is sufficient to compare with each other
the quantities of heat which, during the same instant, cross two
parallel planes.

Let b be the perpendicular distance of these two planes which
we first suppose parallel to the horizontal plane of x and y. Let
m and m be two infinitely near molecules, one of which is above
the first horizontal plane and the other below it : let x, y, z be
the co-ordinates of the first molecule, and x, yf, z those of the
second. In like manner let M and M' denote two infinitely
near 'molecules, separated by the second horizontal plane and
situated, relatively to that plane, in the same manner as m and
m are relatively to the first plane ; that is to say, the co-ordinates
of M are a?, y, z + b, and those of M' are x, y', z + b. It is evident
that the distance mm of the two molecules m and mf is equal
to the distance MM' of the two molecules M and Mf ; further,
let v be the temperature of m, and v' that of m, also let V and
V be the temperatures of M and Mf, it is easy to see that the
two differences v — v and V— V are equal ; in fact, substituting
first the co-ordinates of m and m in the general equation

v — A + ax -f by + cz,
we find v — v = a (x - x) -f b (y — y} + c (z — z},

SECT. VII.] MOVEMENT IN THREE DIMENSIONS. 75

and then substituting the co-ordinates of M and J/', we find also
V— V = a (x — x) + b (y — y) +c(z— /). Now the quantity of
heat which m sends to m depends on the distance mm, which
separates these molecules, and it is proportional to the difference
v — v of their temperatures. This quantity of heat transferred
may be represented by

q(v-v')dt;

the value of the coefficient q depends in some manner on the
distance mm, and on the nature of the substance of which the
solid is formed, dt is the duration of the instant. The quantity
of heat transferred from M to M't or the action of M on M' is
expressed likewise by q (V—V) dt, and the coefficient q is the
same as in the expression q (v — v) dt, since the distance MM' is
equal to mm and the two actions are effected in the same solid :
furthermore V— V is equal to v — v, hence the two actions are
equal.

If we choose two other points n and ri, very near to each
other, which transfer heat across the first horizontal plane, we
shall find in the same manner that their action is equal to that
of two homologous points N and N' which communicate heat
across the second horizontal plane. We conclude then that the
whole quantity of heat which crosses the first plane is equal to
that which crosses the second plane during the same instant.
We should derive the same result from the comparison of two
planes parallel to the plane of x and z, or from the comparison
of two other planes parallel to the plane of y and z. Hence
any part whatever of the solid enclosed between six planes at
right angles, receives through each of its faces as much heat as
it loses through the opposite face ; hence no portion of the solid
can change temperature.

94). From this we see that, across one of the planes in
question, a quantity of heat flows which is the same at all in
stants, and which is also the same for all other parallel sections.

In order to determine the value of this constant flow we
shall compare it with the quantity of heat which flows uniformly
in the most simple case, which has been already discussed. The
case is that of an infinite solid enclosed between two infinite

76 THEORY OF HEAT. [CHAP. I.

planes and maintained in a constant state. We have seen that
the temperatures of the different points of the mass are in this
case represented by the equation v — A + cz ; we proceed to prove
that the uniform flow of heat propagated in the vertical direction
in the infinite solid is equal to that which flows in the same
direction across the prism enclosed by six planes at right angles.
This equality necessarily exists if the coefficient c in the equation
v = A + cz, belonging to the first solid, is the same as the coeffi
cient c in the more general equation v — A + ax + ~by + cz which
represents the state of the prism. In fact, denoting by H a
plane in this prism perpendicular to zt and by m and /JL two
molecules very near to each other, the first of which m is below
the plane H, and the second above this plane, let v be the
temperature of m whose co-ordinates are x, y, z, and w the
temperature of //, whose co-ordinates are x -H a, y + /3. z + 7. Take
a third molecule fi whose co-ordinates are x — a., y — /3, # + y, and
whose temperature may be denoted by w. We see that fju and
fju are on the same horizontal plane, and that the vertical drawn
from the middle point of the line fjup ', which joins these two
points, passes through the point m, so that the distances mjj, and
mfjf are equal. The action of m on ^ or the quantity of heat
which the first of these molecules sends to the other across the
plane H, depends on the difference v - w of their temperatures.
The action of m on p depends in the same manner on the
difference v — w of the temperatures of these molecules, since
the distance of m from fju is the same as that of m from /*' '. Thus,
expressing by q (v — w) the action of m on //, during the unit of
time, we shall have q (v — w) to express the action of m on fjf,
q being a common unknown factor, depending on the distance
nifjb and on the nature of the solid. Hence the sum of the two
actions exerted during unit of time is q (v — w + v — w'}.
If instead of x, y, and zt in the general equation

v = A + ax + by + cz,

we substitute the co-ordinates of m and then those of p and //,
we shall find

t? — w = — act — 6/3 — c%

v — w = + ay. + bft — cy.

SECT. TIL] MOVEMENT IX THREE DIMENSIONS. 77

The sum of the two actions of m on fj, and of m on // is there
fore — 2qcy.

Suppose then that the plane H belongs to the infinite solid
whose temperature equation is v = A + cz, and that we denote
also by mt JJL and p those molecules in this solid whose co
ordinates are x, y, z for the first, x + a, y + /3, z 4- 7 for the second,
and x — a,y — j3,z+y for the third : we shall have, as in the
preceding case, v-w + v-w' = - 2cy. Thus the sum of the two
actions of m on //- and of m on p, is the same in the infinite solid
as in the prism enclosed between the six planes at right angles.

We should obtain a similar result, if we considered the action
of another point n below the plane H on two others v and v ',
situated at the same height above the plane. Hence, the sum
of all the actions of this kind, which are exerted across the plane
H, that is to say the whole quantity of heat which, during unit
of time, passes to the upper side of this surface, by virtue of the
action of very near molecules which it separates, is always the
same in both solids.

95. In the second of these two bodies, that which is bounded
by two infinite planes, and whose temperature equation is
v = A + cz, we know that the quantity of heat which flows during
unit of time across unit of area taken on any horizontal section
whatever is — cK, c being the coefficient of z, and K the specific
conducibility ; hence, the quantity of heat which, in the prism
enclosed between six planes at right angles, crosses during unit
of time, unit of area taken on any horizontal section whatever,
is also - cKy when the linear equation which represents the tem
peratures of the prism is

v = A + ax + by + cz.

In the same way it may be proved that the quantity of heat
which, during unit of time, flows uniformly across unit of area
taken on any section whatever perpendicular to x, is expressed
by - aK, and that the whole quantity which, during unit of time,
crosses unit of area taken on a section perpendicular to y, is
expressed by — bK.

The theorems which we have demonstrated in this and the
two preceding articles, suppose the direct action of heat in the

78 THEORY OF HEAT. [CHAP. I.

interior of the mass to be limited to an extremely small distance,
but they would still be true, if the rays of heat sent out by each
molecule could penetrate directly to a quite appreciable distance,
but it would be necessary in this case, as we have remarked in
Article 70, to suppose that the cause which maintains the tem
peratures of the faces of the solid affects a part extending within
the mass to a finite depth.

. SECTION VIII.
Measure of the movement of heat at a given point of a solid mass.

96. It still remains for us to determine one of the principal
elements of the theory of heat, which consists in defining and in
measuring exactly the quantity of heat which passes through
every point of a solid mass across a plane whose direction is given.

If heat is unequally distributed amongst the molecules of the
same body, the temperatures at ^ any point will vary every instant.
Denoting by t the time which has elapsed, and by v the tem
perature attained after a time t by an infinitely small molecule
whose co-ordinates are oc, y, z ; the variable state of the solid will be
expressed by an equation similar to the following v = F(x, y, z, t).
Suppose the function F to be given, and that consequently we
can determine at every instant the temperature of any point
whatever; imagine that through the point m we draw a hori
zontal plane parallel to that of x and y, and that on this plane
we trace an infinitely small circle «, whose centre is at m ; it is
required to determine what is the quantity of heat which during
the instant dt will pass across the circle a> from the part of the
solid which is below the plane into the part above it.

All points extremely near to the point m and under the plane
exert their action during the infinitely small instant dt, on all
those which are above the plane and extremely near to the point
m, that is to say, each of the points situated on one side of this
plane will send heat to each of those which are situated on the
other side.

We shall consider as positive an action whose effect is to
transport a certain quantity of heat above the plane, and as
negative that which causes heat to pass below the plane. The

SECT. VIII.] MOVEMENT IX A SOLID MASS. 79

sum of all the partial actions which are exerted across the circle
co, that is to say the sum of all the quantities of heat which,
crossing any point whatever of this circle, pass from the part
of the solid below the plane to the part above, compose the flow
whose expression is to be found.

It is easy to imagine that this flow may not be the same
throughout the whole extent of the solid, and that if at another
point m we traced a horizontal circle co equal to the former, the
two quantities of heat which rise above these planes o> and o>'
during the same instant might not be equal : these quantities are
comparable with each other and their ratios are numbers which
may be easily determined.

97. We know already the value of the constant flow for the
case of linear and uniform movement; thus in the solid enclosed be
tween two infinite horizontal planes, one of which is maintained at
the temperature a and the other at the temperature b, the flow of
heat is the same for every part of the mass ; we may regard it as
taking place in the vertical direction only. The value correspond

ing to unit of surface and to unit of time is K ( — —),6 denoting

the perpendicular distance of the two planes, and K the specific
conducibility : the temperatures at the different points of the

solid are expressed by the equation v — a— (— - ) £•

When the problem is that of a solid comprised between six
rectangular planes, pairs of which are parallel, and the tem
peratures at the different points are expressed by the equation

the propagation takes place at the same time along the directions
of x, of y, of z\ the quantity of heat which flows across a definite
portion of a plane parallel to that of x and y is the same through
out the whole extent of the prism ; its value corresponding to unit
of surface, and to unit of time is — cK, in the direction of z, it is
— IK, in the direction of y, and — aK in that of x.

In general the value of the vertical flow in the two cases which
we have just cited, depends only on the coefficient of z and on

the specific conducibility K\ this value is always equal to — K-r- •

80 THEORY OF HEAT. [CHAP. I.

The expression of the quantity of heat which, during the in
stant dt, flows across a horizontal circle infinitely small, whose area
is &>, and passes in this manner from the part of the solid which is
below the plane of the circle to the part above, is, for the two cases

rrdv j,
in question, — K -^- coat.

98. It is easy now to generalise this result and to recognise
that it exists in every case of the varied movement of heat ex
pressed by the equation v = F (x, y, z, t).

Let us in fact denote by x, y, z , the co-ordinates of this point
m, and its actual temperature by v. Let x + f, y + rj, z -f f, be
the co-ordinates of a point JJL infinitely near to the point m, and
whose temperature is w ; f, r\, £ are quantities infinitely small added
to the co-ordinates x , y , z ; they determine the position of
molecules infinitely near to the point m, with respect to three
rectangular axes, whose origin is at m, parallel to the axes of
x, y, and z. Differentiating the equation

0 =/ 0> y> z> 0

and replacing the differentials by f, rj, £ we shall have, to express
the value of w which is equivalent to v + dv, the linear equation

, dv ,. dv dv ^ , m . , dv dv dv f
w = v + j- f + ~j- v + -7- ? ; the coefficients v , -y-, --,-,— i- , are func-
dx dy dz . dx dy dz

tions of x, y, z, t, in which the given and constant values of, y'} z,
which belong to the point m, have been substituted for x, y> z.

Suppose that the same point m belongs also to a solid enclosed
between six rectangular planes, and that the actual temperatures
of the points of this prism, whose dimensions are finite, are ex
pressed by the linear equation w = A + a£ + Irj + c£ ; and that
the molecules situated on the faces which bound the solid are
maintained by some external cause at the temperature which is
assigned to them by the linear equation, f, rj, £ are the rectangular
co-ordinates of a molecule of the prism, whose temperature is wt
referred to three axes whose origin is at m.

This arranged, if we take as the values of the constant coeffi
cients A, a, 6, c, which enter into the equation for the prism^ the

,.,. , dv' dv' dvr , . , , , ,. ,,-p <..

quantities v , -y- , -=- , -=- , which belong to the ditierential eqna-
cLoc dy cLz

tion ; the state of the prism expressed by the equation

SECT. VIII.] MOVEMENT IX A SOLID MASS. 81

, , dv' dv dv „
w = v + -j— £ + -T- *? + -j- ?
ax * dgp cfe

will coincide as nearly as possible with the state of the solid ; that
is to say, all the molecules infinitely near to the point m will have
the same temperature, whether we consider them to be in the solid
or in the prism. This coincidence of the solid and the prism is
quite analogous to that of curved surfaces with the planes which
touch them.

It is evident, from this, that the quantity of heat which flows
in the solid across the circle co, during the instant dt, is the same
as that which flows in the prism across the same circle; for all the
molecules whose actions concur in one effect or the other, have
the same temperature in the two solids. Hence, the flow in

question, in one solid or the other, is expressed by — K -=- wdt.
It would be — K -=- codt, if the circle o>, whose centre is m, were

perpendicular to the axis of y, and — K -^- codt, if this circle were
perpendicular to the axis of x.

The value of the flow which we have just determined varies
in the solid from one point to another, and it varies also with
the time. We might imagine it to have, at all the points of a
unit of surface, the same value as at the point m, and to preserve
this value during unit of time ; the flow would then be expressed

by — K-j- , it would be — K-j- in the direction of y, and — K~
dz, dy dx

in that of x. We shall ordinarily employ in calculation this
value of the flow thus referred to unit of time and to unit of
surface.

99. This theorem serves in general to measure the velocity
with which heat tends to traverse a given point of a plane
situated in any manner whatever in the interior of a solid whose
temperatures vary with the time. Through the given point m,
a perpendicular must be raised upon the plane, and at every
point of this perpendicular ordinates must be drawn to represent
the actual temperatures at its different points. A plane curve
will thus be formed whose axis of abscissse is the perpendicular.

F. H. 6

82 THEORY OF HEAT. [CHAP. I.

The fluxion of the ordinate of this curve, answering to the point
ra, taken with the opposite sign, expresses the velocity with
which heat is transferred across the plane. This fluxion of the
ordinate is known to be the tangent of the angle formed by
the element of the curve with a parallel to the abscissse.

The result which we have just explained is that of which
the most frequent applications have been made in the theory
of heat. We cannot discuss the different problems without
forming a very exact idea of the value of the flow at every point
of a body whose temperatures are variable. It is necessary to
insist on this fundamental notion ; an example which we are
about to refer to will indicate more clearly the use which has
been made of it in analysis.

100. Suppose the different points of a cubic mass, an edge
of which has the length TT, to have unequal actual temperatures
represented by the equation v = cos x cos y cos z. The co
ordinates x, y, z are measured on three rectangular axes, whose
origin is at the centre of the cube, perpendicular to the faces.
The points of the external surface of the solid are at the actual
temperature 0, and it is supposed also that external causes
maintain at all these points the actual temperature 0. On this
hypothesis the body will be cooled more and more, the tem
peratures of all the points situated in the interior of the mass
will vary, and, after an infinite time, they will all attain the
temperature 0 of the surface. Now, we shall prove in the sequel,
that the variable state of this solid is expressed by the equation

v = e~9t cos x cos y cos z,

3/iT

the coefficient g is equal to —* — 71 » -^ *s ^ne specific conduci-

G . I)

bility of the substance of which the solid is formed, D is the
density and G the specific heat ; t is the time elapsed.

We here suppose that the truth of this equation is admitted,
and we proceed to examine the use which may be made of it
to find the quantity of heat which crosses a given plane parallel
to one of the three planes at the right angles.

If, through the point m, whose co-ordinates are x, y, z, we
draw a plane perpendicular to z, we shall find, after the mode

SECT. VIII.] MOVEMENT IN A CUBE. 83

of the preceding article, that the value of the flow, at this point

and across the plane, is — K -j- , or Ke~3t cos x . cos y . sin z. The

clz

quantity of heat which, during the instant dt, crosses an infinitely
small rectangle, situated on this plane, and whose sides are
dx and dy, is

K e* cos x cos y sin z dx dy dt.

Thus the whole heat which, during the instant dt, crosses the
entire area of the same plane, is

K e gf sin z . dt / / cos x cos ydxdy;

the double integral being taken from x = — ^ IT up to x = •= TT,

and from y = — - TT up to y = - TT. We find then for the ex-

— *£

pression of this total heat,

4 A'V'sin^.ok

If then we take the integral with respect to t, from t = 0 to
t = £, we shall find the quantity of heat which has crossed the
same plane since the cooling began up to the actual moment.

This integral is — sin z (1 — e~gt), its value at the surface is

so that after an infinite time the quantity of heat lost through
one of the faces is . The same reasoning being applicable
to each of the six faces, we conclude that the solid has lost by its

complete cooling a total quantity of heat equal to - - or SCD,

*J

since g is equivalent to -^^ . The total heat which is dissipated

C.L/

during the cooling must indeed be independent of the special
conducibility K, which can only influence more or less the
velocity of cooling.

C— 2

84 THEORY OF HEAT. [CH. I. SECT. VIII.

100. A. We may determine in another manner the quantity
of heat which the solid loses during a given time, and this will
serve in some degree to verify the preceding calculation. In
fact, the mass of the rectangular molecule whose dimensions are
dx, dy, dz, is D dx dy dz, consequently the quantity of heat
which must be given to it to bring it from the temperature 0 to
that of boiling water is CD dx dy dz, and if it were required to
raise this molecule to the temperature v, the expenditure of heat
would be v CD dx dy dz.

It follows from this, that in order to find the quantity by
which the heat of the solid, after time t, exceeds that which
it contained at the temperature 0, we must take the mul

tiple integral 1 1 1 v CD dx dy dz, between the limits x = — •= iry

We thus find, on substituting for v its value, that is to say

~9t

e cos x cos y cos z,

that the excess of actual heat over that which belongs to the
temperature 0 is 8 CD (1 — e~gt) ; or, after an infinite time,
8 CD, as we found before.

We have described, in this introduction, all the elements which
it is necessary to know in order to solve different problems
relating to the movement of heat in solid bodies, and we have
given some applications of these principles, in order to shew
the mode of employing them in analysis ; the most important
use which we have been able to make of them, is to deduce
from them the general equations of the propagation of heat,
which is the subject of the next chapter.

Note on Art. 76. The researches of J. D. Forbes on the temperatures of a long
iron bar heated at one end shew conclusively that the conducting power K is not con
stant, but diminishes as the temperature increases. — Transactions of the Eoyal
Society of Edinburgh, Vol. xxiu. pp. 133 — 146 and Vol. xxiv. pp. 73 — 110.

Note on Art. 98. General expressions for the flow of heat within a mass in
which the conductibility varies with the direction of the flow are investigated by
Lame in his Theorie Analytique de la Chaleur, pp. 1 — 8. [A. F.]

CHAPTER II.

EQUATIONS OF THE MOVEMENT OF HEAT.

SECTION I.
Equation of the varied movement of heat in a ring.

101. WE might form the general equations which represent
the movement of heat in solid bodies of any form whatever, and
apply them -to particular cases. But this method would often
involve very complicated calculations which may easily be avoided.
There are several problems which it is preferable to treat in a
special manner by expressing the conditions which are appropriate
to them; we proceed to adopt this course and examine separately
the problems which have been enunciated in the first section of
the introduction ; we will limit ourselves at first to forming the
differential equations, and shall give the integrals of them in the
following chapters.

102. We have already considered the uniform movement of
heat in a prismatic bar of small thickness whose extremity is
immersed in a constant source of heat. This first case offered no
difficulties, since there was no reference except to the permanent
state of the temperatures, and the equation which expresses them
is easily integrated. The following problem requires a more pro
found investigation; its object is to determine the variable state
of a solid ring whose different points have received initial tempe
ratures entirely arbitrary.

The solid ring or armlet is generated by the revolution of
a rectangular section about an axis perpendicular to the plane of

86 THEOKY OF HEAT. [CHAP. II.

the ring (see figure 3), I is the perimeter of the section whose area
*s ^ tne coen<icient h measures the external con-
ducibility, K the internal conducibility, 0 the
specific capacity for heat, D the density. The line
oxos'x" represents the mean circumference of the
armlet, or that line which passes through the
centres of figure of all the sections; the distance
of a section from the origin o is measured by the

arc whose length is x\ R is the radius of the mean circumference.
It is supposed that on account of the small dimensions and of

the form of the section, we may consider the temperature at the

different points of the same section to be equal.

103. Imagine that initial arbitrary temperatures have been
given to the different sections of the armlet, and that the solid is
then exposed to air maintained at the temperature 0, and dis
placed with a constant velocity; the system of temperatures will
continually vary, heat will be propagated within the ring, and
dispersed at the surface: it is required to determine what will be
the state of the solid at any given instant.

Let v be the temperature which the section situated at distance
x will have acquired after a lapse of time t ; v is a certain function
of x and t, into which all the initial temperatures also must enter :
this is the function which is to be discovered.

104. We will consider the movement of heat in an infinitely
small slice, enclosed between a section made at distance x and
another section made at distance x -f dx. The state of this slice
for the duration of one instant is that of an infinite solid termi
nated by two parallel planes maintained at unequal temperatures ;
thus the quantity of heat which flows during this instant dt across
the first section, and passes in this way from the part of the solid
which precedes the slice into the slice itself, is measured according
to the principles established in the introduction, by the product of
four factors, that is to say, the conducibility K, the area of the

section S, the ratio — -=- , and the duration of the instant; its
dx

expression is — KS -j- dt. To determine the quantity of heat

SECT. I.] VARIED MOVEMENT IN A RING. 8?

which escapes from the same slice across the second section, and
passes into the contiguous part of the solid, it is only necessary
to change x into x 4- dx in the preceding expression, or, which is
the same thing, to add to this expression its differential taken
with respect to x ; thus the slice receives through one of its faces

a quantity of heat equal to —KS-j-dt, and loses through the
opposite face a quantity of heat expressed by

Tr. ~ - -, -rr- n , ,

- KS-j- dt - KS T-O dx dt.
dx dx

It acquires therefore by reason of its position a quantity of heat
equal to the difference of the two preceding quantities, that is

KS—ldxdt.
dx?

On the other hand, the same slice, whose external surface is
Idx and whose temperature differs infinitely little from v, allows
a quantity of heat equivalent to hlvdxdt to escape into the air;
during the instant dt\ it follows from this that this infinitely-
small part of the solid retains in reality a quantity of heat

72

represented by K S -^ dx dt - hlv dx dt which makes its tempe-
clx

rature vary. The amount of this change must be examined.

105. The coefficient C expresses how much heat is required
to raise unit of weight of the substance in question from tempe
rature 0 up to temperature 1 ; consequently, multiplying the
volume Sdx of the infinitely small slice by the density Z>, to
obtain its weight, and by C the specific capacity for heat, we shall
have CD Sdx as the quantity of heat which would raise the
volume of the slice from temperature 0 up to temperature 1.
Hence the increase of temperature which results from the addition

J7°

of a quantity of heat equal to KS -^ dx dt — hlv dx dt will be

found by dividing the last quantity by CD Sdx. Denoting there
fore, according to custom, the increase of temperature which takes

place during the instant dt by -,y dt, we shall have the equation

88 THEORY OF HEAT. [CHAP. II.

7/7 TTr) j~Z$. ~~ ~nf)<3 ® vv

CiU \J U UiOC L/X/AJ

We shall explain in the sequel the use which may be made of
this equation to determine the complete solution, and what the
difficulty of the problem consists in; we limit ourselves here to
a remark concerning the permanent state of the armlet.

106. Suppose that, the plane of the ring being horizontal,
sources of heat, each of which exerts a constant action, are placed
below different points m, n, p, q etc. ; heat will be propagated in
the solid, and that which is dissipated through the surface being
incessantly replaced by that which emanates from the sources, the
temperature of every section of the solid will approach more and
more to a stationary value which varies from one section to
another. In order to express by means of equation (b) the law of
the latter temperatures, which would exist of themselves if they
were once established, we must suppose that the quantity v does

not vary with respect to t'} which annuls the term -j-. We thus
have the equation

Ul V fill -I -mif — X\f T7-Q TIT "J^V IfSf

-T~* = ~T7« v> whence v = Me KS + Ne ,

ax AD

M and N being two constants1.

1 This equation is the same as the equation for the steady temperature of a
finite bar heated at one end (Art. 76), except that I here denotes the perimeter of
a section whose area is 8. In the case of the finite bar we can determine two
relations between the constants M and N : for, if V be the temperature at the
source, where # = 0, V—M + N', and if at the end of the bar remote from the source,
where x = L suppose, we make a section at a distance dx from that end, the flow

through this section is, in unit of time, - KS — , and this is equal to the waste

of heat through the periphery and free end of the slice, hv(ldx + S) namely;
hence ultimately, dx vanishing,

=L ^ • *

^ <*!.

IT, irr\ rfjJf1

Cf. Verdet, Conferences de Physique, p. 37. [A. F.]

SECT. I.] STEADY MOVEMENT IN A RING. 89

107. Suppose a portion of the circumference of the ring,
situated between two successive sources of heat, to be divided
into equal parts, and denote by vlt V2, V3, v4, &c., the temperatures
at the points of division whose distances from the origin are
xv xv xv #4> &c-j the relation between v and x will be given by
the preceding equation, after that the two constants have been
determined by means of the two values of v corresponding to

Ju
the sources of heat. Denoting by a the quantity e KS , and

by X the distance x2 — x^ of two consecutive points of division,
we shall have the equations :

whence we derive the following relation — - * = ax + a~A.

We should find a similar result for the three points whose
temperatures are v2, vs, v4, and in general for any three consecutive
points. It follows from this that if we observed the temperatures
v\> vv vs> vv V5 &c- °f several successive points, all situated between
the same two sources m and n and separated by a constant
interval X, we should perceive that any three consecutive tempe
ratures are always such that the sum of the two extremes divided
by the mean gives a constant quotient ax + a~A.

108. If, in the space included between the next two sources of
lieat n and p, the temperatures of other different points separated
by the same interval X were observed, it would still be found that
for any three consecutive points, the sum of the two extreme
temperatures, divided by the mean, gives the same quotient
k*. 4. a-\ The value of this quotient depends neither on the
position nor on the intensity of the sources of heat.

109. Let q be this constant value, we have the equation

Vs $».-«;

we see by this that when the circumference is divided into equal
parts, the temperatures at the points of division, included between

90 THEORY OF HEAT. [CHAP. IT.

two consecutive sources of heat, are represented by the terms of
a recurring series whose scale of relation is composed of two terms
q and — 1.

Experiments have fully confirmed this result. We have ex
posed a metallic ring to the permanent and simultaneous action
of different sources of heat, and we have observed the stationary
temperatures of several points separated by constant intervals; we
always found that the temperatures of any three consecutive
points, not separated by a source of heat, were connected by the
relation in question. Even if the sources of heat be multiplied,
and in whatever manner they be disposed, no change can be

v ~\~ v
effected in the numerical value of the quotient -1- 3 ; it depends

only on the dimensions or on the nature of the ring, and not on
the manner in which that solid is heated.

110. When we have found, by observation, the value of the
constant quotient q or 1 ^ 3 , the value of ax may be derived

from it by means of the equation aA + of A = q. One of the roots
is a\ and other root is a~\ This quantity being determined,

we may derive from it the value of the ratio ^, which is

J\.
o

•j (log a)2. Denoting ax by co, we shall have o>2 — qw + 1 = 0. Thus
I

nr

the ratio of the two conducibilities is found by multiplying —

L

by the square of the hyperbolic logarithm of one of the roots of
the equation o>2 — qa> + 1 = 0, and dividing the product by X2.

SECTION II.

Equation of the varied movement of heat in a solid sphere.

111. A solid homogeneous mass, of the form of a sphere,
having been immersed for an infinite time in a medium main
tained at a permanent temperature 1, is then exposed to air which
is kept at temperature 0, and displaced with constant velocity :
it is required to determine the successive states of the body during
the whole time of the cooling.

SECT. II.] .VARIED MOVEMENT IN A SPHERE. 91

Denote by x the distance of any point whatever from the
centre of the sphere, and by v the temperature of the same point,
after a time t has elapsed ; and suppose, to make the problem
more general, that the initial temperature, common to all points
situated at the distance x from the centre, is different for different
values of x ; which is what would have been the case if the im
mersion had not lasted for an infinite time.

Points of the solid, equally distant from the centre, will not
cease to have a common temperature ; v is thus a function of x
and t. When we suppose t = 0, it is essential that the value of
this function should agree with the initial state which is given,
and which is entirely arbitrary.

112. We shall consider the instantaneous movement of heat
in an infinitely thin shell, bounded by two spherical surfaces whose
radii are x and x + dx: the quantity of heat which, during an
infinitely small instant dt, crosses the lesser surface whose radius
is x, and so passes from that part of the solid which is nearest to
the centre into the spherical shell, is equal to the product of four
factors which are the conducibility K, the duration dt, the extent

^Trx2 of surface, and the ratio -j- , taken with the negative sign ;

it is expressed by — AKirx* -j- dt.

To determine the quantity of heat which flows during the
same instant through the second surface of the same shell, and
passes from this shell into the part of the solid which envelops it,
x must be changed into x + dx, in the preceding expression : that

ci i)

is to say, to the term — ±KTTX* -T- dt must be added the differen
tial of this term taken with respect to x. We thus find

- tKvx* ^dt- IKtrd (x* ^} . dt
dx \ dxj

as the expression of the quantity of heat which leaves the spheri
cal shell across its second surface; and if we subtract this quantity
from that which enters through the first surface, we shall have

xz --} dt. This difference is evidently the quantity of

92 THEORY OF HEAT. [CHAP. II.

heat which accumulates in the intervening shell, and whose effect
is to vary its temperature.

113. The coefficient C denotes the quantity of heat which is
necessary to raise, from temperature 0 to temperature 1, a definite
unit of weight ; D is the weight of unit of volume, ^Trx^dx is the
volume of the intervening layer, differing from it only by a
quantity which may be omitted : hence kjrCDx^dx is the quantity
of heat necessary to raise the intervening shell from temperature
0 to temperature 1. Hence it is requisite to divide the quantity
of heat which accumulates in this shell by 4rjrCDx2dx) and we
shall then find the increase of its temperature v during the time
dt. We thus obtain the equation

Jr d(x2—}
, _ K , \ dxj

~ CD x*dx

v 2 dv\

or -77 = TTT: • I -r-a + - -7- / (c).

5 x dxj ^ '

114. The preceding equation represents the law of the move
ment of heat in the interior of the solid, but the temperatures of
points in the surface are subject also to a special condition which
must be expressed. This condition relative to the state of the
surface may vary according to the nature of the problems dis
cussed : we may suppose for example, that, after having heated
the sphere, and raised all its molecules to the temperature of
boiling water, the cooling is effected by giving to all points in the
surface the temperature 0, and by retaining them at this tem
perature by any external cause whatever. In this case we may
imagine the sphere, whose variable state it is desired to determine,
to be covered by a very thin envelope on which the cooling agency
exerts its action. It may be supposed, 1°, that this infinitely
thin envelope adheres to the solid, that it is of the same substance
as the solid and that it forms a part of it, like the other portions
of the mass ; 2°, that all the molecules of the envelope are sub
jected to temperature Oby a cause always in action which prevents
the temperature from ever being above or below zero. To express
this condition theoretically, the function v, which contains x and t,

SECT. II.] VARIED MOVEMENT IN A SPHERE. 93

must be made to become nul, when we give to x its complete
value X equal to the radius of the sphere, whatever else the value
of t may be. We should then have, on this hypothesis, if we
denote by <f> (x, t) the function of x and t, which expresses the
value of v, the two equations

•jr = -F^ ( -T-2 + - 3- ) , and 6 (X, t) = 0.
dt \jj-J \(zx x cl/jcj

Further, it is necessary that the initial state should be repre
sented by the same function <£ (x, t) : we shall therefore have as a
second condition (/> (x, 0) = 1. Thus the variable state of a solid
sphere on the hypothesis which we have first described will be
represented by a function v, which must satisfy the three preceding
equations. The first is general, and belongs at every instant to
all points of the mass ; the second affects only the molecules at
the surface, and the third belongs only to the initial state.

115. If the solid is being cooled in air, the second equation is
different ; it must then be imagined that the very thin envelope
is maintained by some external cause, in a state such as to pro
duce the escape from the sphere, at every instant, of a quantity of
heat equal to that which the presence of the medium can carry
away from it.

Now the quantity of heat which, during an infinitely small
instant dt, flows within the interior of the solid across the spheri
cal surface situate at distance x, is equal to — 4>K7rxz -^- dt ; and

this general expression is applicable to all values of x. Thus, by
supposing x = X we shall ascertain the quantity of heat which in
the variable state of the sphere would pass across the very thin
envelope which bounds it ; on the other hand, the external surface
of the solid having a variable temperature, which we shall denote
by F, would permit the escape into the air of a quantity of heat
proportional to that temperature, and to the extent of the surface,
which is 4<7rX2. The value of this quantity is 4<h7rX2 Vdt.

To express, as is supposed, that the action of the envelope
supplies the place, at every instant, of that which would result from
the presence of the medium, it is sufficient to equate the quantity

4>JnrX*Vdt to the value which the expression — 4iK TrX* -_,- dt

94 THEORY OF HEAT. [CHAT*. II.

receives when we give to x its complete value X\ hence we obtain

the equation -,- = — -jyV, which must hold when in the functions
dx A

Ct ?J

T and v we put instead of x its value X, which we shall denote
dx

dV

by writing it in the form K ~j- + h V— 0.

doc

116. The value of -=- taken when x = X, must therefore have

dx

a constant ratio — •-+ to the value of v, which corresponds to the

same point. Thus we shall suppose that the external cause of
the cooling determines always the state of the very thin envelope,

C/1J

in such a manner that the value of —, -- which results from this

dx

state, is proportional to the value of v, corresponding to x = X,
and that the constant ratio of these two quantities is — -^ . This
condition being fulfilled by means of some cause always present,
which prevents the extreme value of -y- from being anything else

CLX

but — ^ v, the action of the envelope will take the place of that

of the air.

It is not necessary to suppose the envelope to be extremely
thin, and it will be seen in the sequel that it may have an
indefinite thickness. Here the thickness is considered to be
indefinitely small, so as to fix the attention on the state of the
surface only of the solid.

117. Hence it follows that the three equations which are
required to determine the function $ (x, t} or v are the following,

dn

Tt~~

The first applies to all possible values of x and t ; the second
is satisfied when x = X, whatever be the value of t; and the
third is satisfied when t = 0, whatever be the value of x.

SECT. III.] VARIED MOVEMENT IX A CYLINDER. 95

It might be supposed that in the initial state all the spherical
layers have not the same temperature : which is what would
necessarily happen, if the immersion were imagined not to have
lasted for an indefinite time. In this case, which is more general
than the foregoing, the given function, which expresses the
initial temperature of the molecules situated at distance x from
the centre of the sphere, will be represented by F (x) ; the third
equation will then be replaced by the following, <£ (x, 0) = F (x).

Nothing more remains than a purely analytical problem,
whose solution w7ill be given in one of the following chapters.
It consists in finding the value of v, by means of the general
condition, and the two special conditions to which it is subject.

SECTION III.

Equations of the varied movement of heat in a solid cylinder.

118. A solid cylinder of infinite length, whose side is per
pendicular -to its circular base, having been wholly immersed
in a liquid whose temperature is uniform, has been gradually
heated, in such a manner that all points equally distant from
the axis have acquired the same temperature ; it is then exposed
to a current of colder air ; it is required to determine the
temperatures of the different layers, after a given time.

x denotes the radius of a cylindrical surface, all of whose
points are equally distant from the axis ; X is the radius of
the cylinder ; v is the temperature which points of the solid,
situated at distance x from the axis, must have after the lapse
of a time denoted by t, since the beginning of the cooling.
Thus v is a function of x and t, and if in it t be made equal to
0, the function of x which arises from this must necessarily satisfy
the initial state, which is arbitrary.

119. Consider the movement of heat in an infinitely thin
portion of the cylinder, included between the surface whose
radius is x, and that whose radius is x + dx. The quantity of
heat which this portion receives during the instant dty from the
part of the solid which it envelops, that is to say, the quantity
which during the same time crosses the cylindrical surface

96 THEORY OF HEAT. [CHAP. II.

whose radius is x, and whose length is supposed to be equal
to unity, is expressed by

dx

To find the quantity of heat which, crossing the second surface
whose radius is x + dx, passes from the infinitely thin shell into
the part of the solid which envelops it, we must, in the foregoing
expression, change x into x + dx, or, which is the same thing,
add to the term

— 2K7TX ys- dt,

dx

the differential of this term, taken with respect to x. Hence
the difference of the heat received and the heat lost, or the
quantity of heat which accumulating in the infinitely thin shell
determines the changes of temperature, is the same differential
taken with the opposite sign, or

*&.«.*(.*);

on the other hand, the volume of this intervening shell is Qirxdx,
and ZCDjrxdx expresses the quantity of heat required to raise
it from the temperature 0 to the temperature 1, C being the
specific heat, and D the density. Hence the quotient

~

dx

ZCDwxdx

is the increment which the temperature receives during the
instant dt. Whence we obtain the equation

£k - K (^ ldJL\ • * T !

dt CD \da? x dx) ' \

120. The quantity of heat which, during the instant dt,
crosses the cylindrical surface whose radius is xt being expressed

in general by 2Kirx -j- dt, we shall find that quantity which

escapes during the same time from the surface of the solid, by
making x = X in the foregoing value; on the other hand, the

SECT. IV.] STEADY MOVEMENT IN A PRI-M. 97

same quantity, dispersed into the air, is, by the principle of the
communication of heat, equal to %7rXhvJt ; we must therefore

have at the surface the definite equation —K-j- =hv. The

nature of these equations is explained at greater length, either
in the articles which refer to the sphere, or in those wherein the
general equations have been given for a body of any form what
ever. The function t? which represents the movement of heat in
an infinite cylinder must therefore satisfy, 1st, the general equa-

- dv K (tfv 1 dv\ , . .

tion ~r — ~^T} [TJ ~*~ ~ J~) » wnicn ^PP^es whatever x and t may

be; 2nd, the definite equation -^ v -f -j- = 0, which is true, whatever

the variable t may be, when x — X; 3rd, the definite equation
v = F(x). The last condition must be satisfied by all values
of r, when t is made equal to 0, whatever the variable x may
be. The arbitrary function F (x) is supposed to be known ; it
corresponds to the initial state.

SECTION IV.

Equations of the uniform movement of heat in a solid prism
of infinite length.

121. A prismatic bar is immersed at one extremity in a
constant source of heat which maintains that extremity at the
temperature A ; the rest of the bar, whose length is infinite,
continues to be exposed to a uniform current of atmospheric air
maintained at temperature 0; it is required to determine the
highest temperature which a given point of the bar can acquire.

The problem differs from that of Article 73, since we now W
take into consideration all the dimensions of the solid, which is
necessary in order to obtain an exact solution.

We are led, indeed, to suppose that in a bar of very small
thickness all points of the same section would acquire sensibly
equal temperatures ; but some uncertainty may rest on the
results of this hypothesis. It is therefore preferable to solve the
problem rigorously, and then to examine, by analysis, up to what
point, and in what cases, we are justified in considering the
temperatures of different points of the same section to be equal.
F. H. 7

98 THEORY OF HEAT. [CHAP. II.

122. The section made at right angles to the length of the
bar, is a square whose side is 2f, the axis of the bar is the axis
of x, and the origin is at the extremity A. The three rectangular
co-ordinates of a point of the bar are xt y, z, and v denotes the
fixed temperature at the same point.

The problem consists in determining the temperatures which
must be assigned to different points of the bar, in order that
they may continue to exist without any change, so long as the
extreme surface A, which communicates with the source of heat,
remains subject, at all its points, to the permanent tempera
ture A ; thus v is a function of xt y, and z.

123. Consider the movement of heat in a prismatic molecule,
enclosed between six planes perpendicular to the three axes
of x, y, and z. The first three planes pass through the point m
whose co-ordinates are x, y, z, and the others pass through the
point m whose co-ordinates are x -f dx, y + dy, z-\- dz.

To find what quantity of heat enters the molecule during
unit of time across the first plane passing through the point m
and perpendicular to xt we must remember that the extent of the
surface of the molecule on this plane is dydz, and that the flow
across this area is, according to the theorem of Article 98, equal

to — K — ; thus the molecule receives across the rectangle dydz
dx

passing through the point m a quantity of heat expressed by
'z -j- . To find the quantity of heat which crosses the

opposite face, and escapes from the molecule, we must substitute,
in the preceding expression, x + dx for x, or, which is the same
thing, add to this expression its differential taken with respect
to x only; whence we conclude that the molecule loses, at its
second face perpendicular to x, a quantity of heat equal to

dv fdv\

— A dydz —, -- A dndzd -r- ;
9 dx \dxj '

we must therefore subtract this from that which enters at the
opposite face ; the differences of these two quantities is

tr j j j fdv\

A dydz a I -j- 1 , or, A a x dyd
\ctx/

d2v
z -=-^
dx

SECT. IV.] STEADY MOVEMENT IN A PRISM. 9D

this expresses the quantity of heat accumulated in the molecule
in consequence of the propagation in direction of x ; which ac
cumulated heat would make the temperature of the molecule
vary, if it were not balanced by that which is lost in some other
direction.

It is found in the same manner that a quantity of heat equal

to — Kdz dx -T- enters the molecule across the plane passing

through the point m perpendicular to y, and that the quantity
which escapes at the opposite face is

— Kdzdx -j -- Kdzdx d ( -T- ) ,
dy \dy)

the last differential being taken with respect to y only. Hence

the difference of the two quantities, or Kdxdydz j-$, expresses

dy

the quantity of heat which the molecule acquires, in consequence
of the propagation in direction of y.

Lastly, it is proved in the same manner that the molecule
acquires, in consequence of the propagation in direction of zt

a quantity of heat equal to Kdxdydz-j-j. Now, in order that

dz

there may be no change of temperature, it is necessary for the
molecule to retain as much heat as it contained at first, so that
the heat it acquires in one direction must baknce that
loses in another. Hence the sum of the three quanti
acquired must be nothing; thus we form the equation

d2v cPv tfv _
da?d** dzz~

first, so that
hat which it
ities of heat

124 It remains now to express the conditions relative to the
surface. If we suppose the point m to belong to one of the faces
of the prismatic bar, and the face to be perpendicular to z, we
see that the rectangle dxdy, during unit of time, permits a
quantity of heat equal to Vh dx dy to escape into the air,
V denoting the temperature of the point m of the surface, namely
what <f> (x, y, z] the function sought becomes when z is made
equal to I, half the dimension of the prism. On the other hand,
the quantity of heat which, by virtue of the action of the

7—2

100 THEORY OF HEAT. [CHAP. II.

molecules, during unit of time, traverses an infinitely small surface
G>, situated within the prism, perpendicular to zy is equal to

— Kco-j-, according to the theorems quoted above. This ex-

pression is general, and applying it to points for which the co
ordinate z has its complete value I, we conclude from it that the
quantity of heat which traverses the rectangle dx dy taken at the

surface is - Kdxdy-j-, giving to z in the function -7- its com
plete value I. Hence the two quantities — Kdxdy-j-, and

CLZ

h dx dy v, must be equal, in order that the action of the molecules
may agree with that of the medium. This equality must also

exist when we give to z in the functions -y- and v the value — I,

dz

which it has at the face opposite to that first considered. Further,
the quantity of heat which crosses an infinitely small surface co,

perpendicular to the axis of y, being —Kco-j-, it follows that
that which flows across a rectangle dz dx taken on a face of the

(i rJ

prism perpendicular to y is - K dz dx -=- , giving to y in the

•J

function -y- its complete value I. Now this rectangle dz dx
dy

permits a quantity of heat expressed by hv dx dy to escape into
the air; the equation hv = — K^- becomes therefore necessary,

t/

r/?j

when y is made equal to I or — I in the functions v and -=- .

dy

125. The value of the function v must by hypothesis be
equal to A, when we suppose • a? = 0, whatever be 'the values of
y and z. Thus the required function v is determined by the
following conditions: 1st, for all values of x} y, z, it satisfies the
general equation

d^v d*v d*v _

dtf + dy*+~dz*~

2nd, it satisfies the equation y^w + -r- = 0, when y is equal to

SECT. V.] VARIED MOVEMENT IN A CUBE8. 10 T

I or — I, whatever x and z may be, or satisfies* the equation
-pV + ^- = 0, when z is equal to I or — I, whatever x and y may

be ; 3rd, it satisfies the equation v = A, when x = 0, whatever
y and z may be.

SECTION Y.
Equations of the varied movement of heat in a solid cule.

126. A solid in the form of a cube, all of whose points have
acquired the same temperature, is placed in a uniform current of
atmospheric air, maintained at temperature 0. It is required to
determine the successive states of the body during the whole
time of the cooling.

The centre of the cube is taken as the origin of rectangular
coordinates; the three perpendiculars dropped from this point on
the faces, are the axes of x, y, and z ; 21 is the side of the cube,
v is the temperature to which a point whose coordinates are
x, y} z, is lowered after the time t has elapsed since the com
mencement of the cooling : the problem consists in determining
the function v, which depends on x, y, z and t.

127. To form the general equation which v must satisfy,
we must ascertain what change of temperature an infinitely
small portion of the solid must experience during the instant
dt, by virtue of the action of the molecules which are extremely
near to it. We consider then a prismatic molecule enclosed
between six planes at right angles; the first three pass through
the point m, whose co-ordinates are x, y, z, and the three others,
through the point m', whose co-ordinates are

x + dx, y + dy, z + dz.

The quantity of heat which during the instant dt passes into
the molecule across the first rectangle dy dz perpendicular to x,

is — Kdy dz -T- dt, and that which escapes in the same time from

the molecule, through the opposite face, is found by writing
x-}- dx in place of x in the preceding expression, it is

- Kdy ^ ( -y-J dt.— Kdy dzd(-^\ dt,

102 THEORY OF HEAT. [CHAP. II.

the differential being taken with respect to x only. The quantity
of heat which during the instant dt enters the molecule, across
the first rectangle dz dx perpendicular to the axis of y, is

— Kdzdx--.~dt, and that which escapes from the molecule during
the same instant, by the opposite face, is

— Kdz dx 4- dt — Kdz dx d ( -y- ) dt,
ay \dyJ

the differential being taken with respect to y only. The quantity
of heat which the molecule receives during the instant dt, through

its lower face, perpendicular to the axis of z, is —Kdxdy-j-dt,

dz

and that which it loses through the opposite face is
~Kdxdy^dt-Kdxdyd(~^dt,

the differential being taken with respect to z only.

The sum of all the quantities of heat which escape from the
molecule must now be deducted from the sum of the quantities
which it receives, and the difference is that which determines its
increase of temperature during the instant: this difference is

Kdij dz d -. dt + Kdz dx d dt + K dx dy d dt,

128. If the quantity which has just been found be divided by
that which is necessary to raise the molecule from the temperature
0 to the temperature 1, the increase of temperature which is
effected during the instant dt will become known. Now, the
latter quantity is CD dx dy dz : for C denotes the capacity of
the substance for heat; D its density, and dxdydz the volume
of the molecule. The movement of heat in the interior of the
solid is therefore expressed by the equation

dv K fd^v d^v d*v\

. — / j_ . i __ ( fj \

7 t ~" f°1 7~\ I 7 *2 * I 2 I 7 • I ••••••••••••• I W Ji

dt CD \dx' dy* dz J ^ '

SECT. V.] VAIIIED MOVEMENT IX A CUBE. 103

129. It remains to form the equations which relate to the
state of the surface, which presents no difficulty, in accordance
with the principles which we have established. In fact, the
quantity of heat Avhich, during the instant dt: crosses the rectangle

dz dy, traced on a plane perpendicular to x} is — K dy dz -v- dt.

This result, which applies to all points of the solid, ought to hold
when, the value of x is equal to I, half the thickness of the prism.
In this case, the rectangle dyds being situated at the surface, the
quantity of heat which crosses it, and is dispersed into the air
during the instant dt, is expressed by hvdydz dt, we ought there
fore to have, when x = l} the equation hv = —K-j-. This con-

CL*k

dition must also be satisfied when x = — I.

It will be found also that, the quantity of heat which crosses
the rectangle dz dx situated on a plane perpendicular to the axis

of y being in general — Kdz dx -j- , and that which escapes at the
surface into "the air across the same rectangle being hvdzdxdt,
we must have the equation hu + K-j- = Q, when y — l or — L

U

Lastly, we obtain in like manner the definite equation

dz

which is satisfied when z = I or — L

130. The function sought, which expresses the varied move
ment of heat in the interior of a solid of cubic form, must therefore
be determined by the following conditions :

1st. It satisfies the general equation

2nd. It satisfies the three definite equations

, ,

dx ay

which hold when x= ±1, y = ±1, z= ±1;

104 THEORY OF HEAT. [CHAP. II.

3rd. If in the function v which contains x, y, z, t, we make
t — 0, whatever be the values of x, y, and z, we ought to have,
according to hypothesis, v = A, which is the initial and common
value of the temperature.

131. The equation arrived at in the preceding problem
represents the movement of heat in the interior of all solids.
Whatever, in fact, the form of the body may be, it is evident that,
by decomposing it into prismatic molecules, we shall obtain this
result. We may therefore limit ourselves to demonstrating in
this manner the equation of the propagation of heat. But in
order to make the exhibition of principles more complete, and
that we may collect into a small number of consecutive articles
the theorems which serve to establish the general equation of the
propagation of heat in the interior of solids, and the equations
which relate to the state of the surface, we shall proceed, in the
two following sections, to the investigation of these equations,
independently of any particular problem, and without reverting
to the elementary propositions which we have explained in the
introduction.

SECTION VI.

General equation of the propagation of heat in the interior of solids.

132. THEOREM I. If the different points of a homogeneous
solid mass, enclosed between six planes at right angles, have actual
temperatures determined by the linear equation

v = A — ax — by — cz, (a),

and if the molecules situated at the external surface on the six
planes which bound the prism are maintained, by any cause what
ever, at the temperature expressed by the equation (a) : all the
molecules situated in the interior of the mass will of themselves
retain their actual temperatures, so that there will be no change in
the state of the prism.

v denotes the actual temperature of the point whose co
ordinates are x, y, z ; A, a, b, c, are constant coefficients.

To prove this proposition, consider in the solid any three
points whatever wJ//z, situated on the same straight line m^,

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 105

which the point M divides into two equal parts ; denote by
x, y, z the co-ordinates of the point Mt and its temperature by
v, the co-ordinates of the point p by x + a, y + /3, z + y, and its
temperature by w, the co-ordinates of the point m by as — a, y — fi,
z — y, and its temperature by ut we shall have

v = A — ax — ly — cz,

whence we conclude that,

v — w = az + 6/3 + cy, and u — v = az + b/3 + cy ;
therefore v — w = u — v.

Now the quantity of heat which one point receives from
another depends on the distance between the two points and
on the difference of their temperatures. Hence the action of
the point M on the point //, is equal to the action of m on M;
thus the point M receives as much heat from m as it gives up
to the point p.

We obtain the same result, whatever be the direction and
magnitude of the line which passes through the point J/, and
is divided into two equal parts. Hence it is impossible for this
point to change its temperature, for it receives from all parts
as much heat as it gives up.

The same reasoning applies to all other points ; hence no
change can happen in the state of the solid.

133. COROLLARY I. A solid being enclosed between two
infinite parallel planes A and B, if the actual temperature of
its different points is supposed to be expressed by the equation
v = l—z, and the two planes which bound it are maintained
by any cause whatever, A at the temperature 1, and B at the
temperature 0 ; this particular case will then be included in
the preceding lemma, if we make A=l, a = 0, & = 0, c = 1.

134. COROLLARY II. If in the interior of the same solid
we imagine a plane M parallel to those which bound it, we see
that a certain quantity of heat flows across this plane during
unit of time ; for two very near points, such as m and n, one

106 THEORY OF HEAT. [CHAP. II.

of which is below the plane and the other above it, are unequally
heated; the first, whose temperature is highest, must therefore
send to the second, during each instant, a certain quantity of heat
which, in some cases, may be very small, and even insensible,
according to the nature of the body and the distance of the two
molecules.

The same is true for any two other points whatever separated
by the plane. That which, is most heated sends to the other
a certain quantity of heat, and the sum of these partial actions,
or of all the quantities of heat sent across the plane, composes
a continual flow whose value does not change, since all the
molecules preserve their temperatures. It is easy to prove that
this floiv, or the quantity of heat which crosses the plane M during
the unit of time, is equivalent to that luhich crosses, during the same
time, another plane N parallel to the first. In fact, the part of
the mass which is enclosed between the two surfaces M and
N will receive continually, across the plane M, as much heat
as it loses across the plane N. If the quantity of heat, which
in passing the plane M enters the part of the mass which is
considered, were not equal to that which escapes by the opposite
surface N, the solid enclosed between the two surfaces would
acquire fresh heat, or would lose a part of that which it has,
and its temperatures would not be constant; which is contrary to
the preceding lemma.

135. The measure of the specific conducibility of a given
substance is taken to be the quantity of heat which, in an infinite
solid, formed of this substance, and enclosed between two parallel
planes, flows during unit of time across unit of surface, taken
on any intermediate plane whatever, parallel to the external
planes, the distance between which is equal to unit of length,
one of them being maintained at temperature 1, and the other
at temperature 0. This constant flow of the heat which crosses
the whole extent of the prism is denoted by the coefficient K,
and is the measure of the conducibility.

136. LEMMA. If we suppose all the temperatures of the solid in
question under the preceding article, to be multiplied by any number
whatever g, so that the equation of temperatures is v = g — gz,
instead of bsing v = 1 — z, and if the two external planes are main-

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 107

tained, one at the temperature g, and the other at temperature 0,
the constant flow of heat, in this second hypothesis, or the quantity
which during unit of time crosses unit of surface taken on an
intermediate plane parallel to the bases, is equal to the product
of the first flow multiplied by g.

In fact, since all the temperatures have been increased in
the ratio of 1 to g, the differences of the temperatures of any
two points whatever m and //., are increased in the same ratio.
Hence, according to the principle of the communication of heat,
in order to ascertain the quantity of heat which in sends to ^
on the second hypothesis, we must multiply by g the quantity
which the same point m sends to (JL on the first hypothesis.
The same would be true for any two other points whatever.
Now, the quantity of heat which crosses a plane M results from
the sum of all the actions which the points m, m', m"j m", etc.,
situated on the same side of the plane, exert on the points //.,
//, fju', fj!"} etc., situated on the other side. Hence, if in the first
hypothesis the constant flow is denoted by K} it will be equal to
gK, wrhen we have multiplied all the temperatures by g.

137. THEOREM II. In a prism whose constant temperatures
are expressed by the equation v = A — ax- by — cz, and which
is bounded by six planes at right angles all of whose points are
maintained at constant temperatures determined by the preceding
equation, the quantity of heat which, during unit of time, crosses
unit of surface taken on any intermediate plane whatever perpen
dicular to z, is the same as the constant flow in a solid of the
same substance would be, if enclosed between two infinite parallel
planes, and for which the equation of constant temperatures is
v = c — cz.

To prove this, let us consider in the prism, and also in the
infinite solid, two extremely near points m and p, separated

Fig. 4.

r

m h

by the plane M perpendicular to the axis of z ; ^ being above
the plane, and m below it (see fig. 4), and above the same plane

108 THEORY OF HEAT. [CHAP. II.

let us take a point m such that the perpendicular dropped from
the point //, on the plane may also be perpendicular to the
distance mm at its middle point h. Denote by x, y, z + h, the
co-ordinates of the point //,, whose temperature is w, by x — a, y — /3,
z, the co-ordinates of m, whose temperature is v, and by a? -fa,
y + {3, z, the co-ordinates of m', whose temperature is v.

The action of m on (JL, or the quantity of heat which m sends
to jju during a certain time, may be expressed by q(v — w). The
factor q depends on the distance nip, and on the nature of the
mass. The action of m' on //, will therefore be expressed by
q (v — w) ; and the factor q is the same as in the preceding
expression; hence the sum of the two actions of m on ft, and
of m on //-, or the quantity of heat which //, receives from m and
from m, is expressed by

q ( v — w -f v — w}.
Now, if the points m, p, m belong to the prism, we have

w — A — ax — by — c (z -f h), v = A — a (x — a) — b (y — /3) — cz,
and v = A - a (x + a) - 6 (y + /3) - cz ;

and if the same points belonged to an infinite solid, we should
have, by hypothesis,

w = c — c(z+li)y v = c — cz, and v = c — cz.
In the first case, we find

q (v — w + v — w) = 2qch,

and, in the second case, we still have the same result. Hence
the quantity of heat which //, receives from m and from m on
the first hypothesis, when the equation of constant temperatures
is v = A — ax — by — cz, is equivalent to the quantity of heat
which p receives from m and from m when the equation of
constant temperatures is v = c — cz.

The same conclusion might be drawn with respect to any three
other points whatever m, /// ', m", provided that the second // be
placed at equal distances from the other two, and the altitude of
the isosceles triangle m /jf m" be parallel to z. Now, the quantity
of heat which crosses any plane whatever M, results from the sum
of the actions which all the points m, m', in", in" etc., situated on

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 109

one side of this plane, exert on all the points /JL, //, /z", p" ', etc
situated on the other side : hence the constant flow, which, during
unit of time, crosses a definite part of the plane M in the infinite
solid, is equal to the quantity of heat which flows in the same time
across the same portion of the plane H in the prism, all of whose
temperatures are expressed by the equation

v = A — ax — by - cz.

138. COROLLARY. The flow has the value cK in the infinite
solid, when the part of the plane which it crosses has unit of

surface. In the pi~ism also it has the same value cK or — K -7- .

It is proved in the same manner, that the constant flow which takes
place, during unit of time, in the. same prism across unit of surf ace t
on any plane whatever perpendicular to y, is equal to

dv
bK or — K 3- :

<ty

and that which crosses a plane perpendicular to x lias the value

-.

dx

139. The propositions which we have proved in the preceding
articles apply also to the case in which the instantaneous action of
a molecule is exerted in the interior of the mass up to an appre
ciable distance. In this case, we must suppose that the cause
which maintains the external layers of the body in the state
expressed by the linear equation, affects the mass up to a finite
depth. All observation concurs to prove that in solids and liquids
the distance in question is extremely small.

140. THEOREM III. If the temperatures at the points of a
solid are expressed by the equation v = f (x, y, z, t), in which
a?, y, z are the co-ordinates of a molecule whose temperature is
equal to v after the lapse of a time t; the flow of heat which
crosses part of a plane traced in the solid, perpendicular to one of
the three axes, is no longer constant ; its value is different for
different parts of the plane, and it varies also with the time. This
variable quantity may be determined by analysis.

110 THEORY OF HEAT. [CHAP. II.

Let w be an infinitely small circle whose centre coincides with
the point m of the solid, and whose plane is perpendicular to the
vertical co-ordinate z ; during the instant dt there will flow across
this circle a certain quantity of heat which will pass from the
part of the circle below the plane of the circle into the upper
part. This flow is composed of all the rays of heat which depart
from a lower point arid arrive at an upper point, by crossing
a point of the small surface w. We proceed to shew that the

dv
expression of the value of the flow is — K -7- &>dt.

Let us denote by x, y, z the coordinates of the point m whose
temperature is v ; and suppose all the other molecules to be
referred to this point in chosen as the origin of new axes parallel
to the former axes : let f, 77, f, be the three co-ordinates of a point
referred to the origin m ; in order to express the actual temperature
w of a molecule infinitely near to m, we shall have the linear
equation

, ,. dv dv . dv
w-v + £-r+i7-7— +£-,-.
* dx ' dy dz

The coefficients t/, j-n. -7—, -r- are the values which are found
dx dy dz

by substituting in the functions v,—j-, -j- , -T-, for the variables

x, y z, the constant quantities x r, y, z, which measure the dis
tances of the point m from the first three axes of x, y, and z.

Suppose now that the point m is also an internal molecule of
a rectangular prism, enclosed between six planes perpendicular to
the three axes whose origin is m ; that w the actual temperature of
each molecule of this prism, whose dimensions are finite, is ex
pressed by the linear equation w = A + a% + brj + c£ and that the
six faces which bound the prism are maintained at the fixed tem
peratures which the last equation assigns to them. The state of
the internal molecules will also be permanent, and a quantity of
heat measured by the expression —Kcwdt will flow during the
instant dt across the circle &>.

This arranged, if we take as the values of the constants

7 xi ,-,- • dv dv dv' ,, /» j c ,1

A, a, 6, c, the quantities v , -5— , -y- , -j-t the fixed state of the

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. Ill

prisrn will be expressed by the equation

, dv' dv dv „

w = v +-T-£+-7-^+ JT~?I

dx * dy dz

Thus the molecules infinitely near to the point m will have,
during the instant dt, the same actual temperature in the solid
whose state is variable, and in the prism whose state is constant.
Hence the flow which exists at the point m, during the instant dt,
across the infinitely small circle &>, is the same in either solid ; it

is therefore expressed by — K -7— codt.

CL2

From this we derive the following proposition

If in a solid whose internal temperatures vary with the time, by
virtue of the action of the molecules, we trace any straight line what
ever, and erect (see fig. o), at the different points of this line, the
ordinates pm of a plane curve equal to the temperatures of these
points taken at the same moment; the flow of heat, at each point p
of the straight line, will be proportional to the tangent of the angle
a. which the element of the curve makes with the parallel to the
alscissw ; that is to say, if at the point p we place the centre of an

Fig. 5.

infinitely small circle o> perpendicular to the line, the quantity of
heat which has flowed during the instant dt, across this circle, in
the direction in which the abscissae op increase, will be measured
by the product of four factors, which are, the tangent of the angle
a, a constant coefficient K, the area o> of the circle, and the dura
tion dt of the instant.

141. COROLLARY. If we represent by e the abscissa of this
curve or the distance of a point p of the straight line from a

112 THEORY OF HEAT. [CHAP. II.

fixed point o, and by v the ordinate which represents the tem
perature of the point p, v will vary with the distance e and
will be a certain function /(e) of that distance; the quantity
of heat which would flow across the circle o>, placed at the

point p perpendicular to the line, will be — K -=- wdt, or

-Kf'(e)a>dt,
denoting the function \/ by/'(e).

QJ€.

We may express this result in the following manner, which
facilitates its application.

To obtain the actual flow of heat at a point p of a straight
line drawn in a solid, whose temperatures vary by action of the
molecules, we must divide the difference of the temperatures at
two points infinitely near to the point p by the distance between
these points. The flow is proportional to the quotient.

142. THEOHEM IV. From the preceding Theorems it is
easy to deduce the general equations of the propagation of heat.

Suppose the different points of a homogeneous solid of any
form whatever, to have received initial temperatures which vary
successively by the effect of the mutual action of the molecules,
and suppose the equation v = f (x, y, z, t) to represent the successive
states of the solid, it may now be shewn that v a function of four
variables necessarily satisfies the equation

dy K_ /d2 v dV dV\
dt " CD Vdx2 + dy* + dzV '

In fact, let us consider the movement of heat in a molecule
enclosed between six planes at right angles to the axes of x, y,
and z\ the first three of these planes pass through the point
m whose coordinates are x, y, z, the other three pass through
the point m, whose coordinates are x + dx, y + dy,z + dz.

During the instant dt, the molecule receives, across the
lower rectangle dxdy, which passes through the point m, a

quantity of heat equal to — K dx dy -=- dt. To obtain the quantity

which escapes from the molecule by the opposite face, it is
sufficient to change z into z -f dz in the preceding expression,

SECT. VI.] GENEKAL EQUATIONS OF PROPAGATION. 113

that is to say, to add to this expression its own differential taken
with respect to z only ; we then have

— Kdx dtj -y- dt —Kdx d u — ^ — dz
J dz * dz

as the value of the quantity which escapes across the upper
rectangle. The same molecule receives also across the first
rectangle dz dx which passes through the point m, a quantity

of heat equal to — K-j- dz dx dt ; and if we add to this ex
pression its owrn differential taken with respect to y only, we
find that the quantity which escapes across the opposite face
dz dx is expressed by

— K-j- dz dx dt — K . ^ dy dz dx dt.
y y

Lastly, the molecule receives through the first rectangle dy dz
a quantity of heat equal to — K -y- dy dz dt, and that which it

CiX

loses across the opposite rectangle which passes through m' is
expressed by

,^ 777 -rr dX 7777

—K-r dy dzdt—K — -r — dx dy dz dt.

We must now take the sum of the quantities of heat which
the molecule receives and subtract from it the sum of those
which it loses. Hence it appears that during the instant dt,
a total quantity of heat equal to

accumulates in the interior of the molecule. It remains only
to obtain the increase of temperature which must result from
this addition of heat.

D being the density of the solid, or the weight of unit of

volume, and C the specific capacity, or the quantity of heat

which raises the unit of weight from the temperature 0 to the

temperature 1 ; the product CDdxdydz expresses the quantity

F. H. 8

ll4 THEORY OF HEAT. [CHAP. II.

of heat required to raise from 0 to 1 the molecule whose volume
is dx dydz. Hence dividing by this product the quantity of
heat which the molecule has just acquired, we shall have its
increase of temperature. Thus we obtain the general equation

^ - J^ (^ JL ^ + &1

which is the equation of the propagation of heat in the interior
of all solid bodies.

143. Independently of this equation the system of tempera
tures is often subject to several definite conditions, of which no
general expression can be given, since they depend on the nature
of the problem.

If the dimensions of the mass in which heat is propagated are
finite, and if the surface is maintained by some special cause in a
given state ; for example, if all its points retain, by virtue of that
cause, the constant temperature 0, we shall have, denoting the
unknown function v by (f> (x, y, z, t}, the equation of condition
(j> (x, y, 2, t) = 0 ; which must be satisfied by all values of x, y, z
which belong to points of the external surface, whatever be the
value of t. Further, if we suppose the initial temperatures of the
body to be expressed by the known function F (x, y, z), we have
also the equation <f> (x, y, z, 0) = F (x, y, z) ; the condition ex
pressed by this equation must be fulfilled by all values of the
co-ordinates x, y} z which belong to any point whatever of the
solid.

144. Instead of submitting the surface of the body to a con
stant temperature, we may suppose the temperature not to be
the same at different points of the surface, and that it varies with
the time according to a given law ; which is what takes place in
the problem of terrestrial temperature. In this case the equation
relative to the surface contains the variable t.

145. In order to examine by itself, and from a very general
point of view, the problem of the propagation of heat, the solid
whose initial state is given must be supposed to have all its
dimensions infinite; no special condition disturbs then the dif-

SECT. VII.] GENERAL SURFACE EQUATION. 115

fusion of heat, and the law to which this principle is submitted
becomes more manifest ; it is expressed by the general equation

dt ~ CD

to which must be added that which relates to the initial arbitrary
state of the solid.

Suppose the initial temperature of a molecule, whose co
ordinates are x, y, z} to be a known function F(xt y, z}y and denote
the unknown value v by <f> (x, y, z, t), we shall have the definite
equation <f> (as, y, z, 0) = F (x, y, 2) ; thus the problem is reduced to
the integration of the general equation (A) in such a manner that
it may agree, when the time is zero, with the equation which con
tains the arbitrary function F.

SECTION VII.

General equation relative to the surface.

146. If the solid has a definite form, and if its original heat
is dispersed gradually into atmospheric air maintained at a con
stant temperature, a third condition relative to the state of the
surface must be added to the general equation (A) and to that
which represents the initial state.

We proceed to examine, in the following articles, the nature of
the equation which expresses this third condition.

Consider the variable state of a solid whose heat is dispersed
into air, maintained at the fixed temperature 0. Let o> be an
infinitely small part of the external surface, and p a point of &>,
through which a normal to the surface is drawn ; different points
of this line have at the same instant different temperatures.

Let v be the actual temperature of the point p,, taken at a
definite instant, and w the corresponding temperature of a point v
of the solid taken on the normal, and distant from //, by an in
finitely small quantity a. Denote by x, y, z the co-ordinates of
the point p, and those of the point v by x + &£, y + &y, z + Sz ;
let/ (x, y, z) = 0 be the known equation to the surface of the solid,
and v = </> (x, y, z, f) the general equation which ought to give the

8—2

116 THEORY OF HEAT. [CHAP. II.

value of v as a function of the four variables x, y, z, t. Differen
tiating the equation f(x, y, z) = 0, we shall have

mdx 4- ndy -\-pdz — 0 ;

m, n, p being functions of x, y, z.

It follows from the corollary enunciated in Article 141, that
the flow in direction of the normal, or the quantity of heat which
during the instant dt would cross the surface «, if it were placed
at any point whatever of this line, at right angles to its direction,
is proportional to the quotient which is obtained by dividing the
difference of temperature of two points infinitely near by their
distance. Hence the expression for the flow at the end of the
normal is

T^w — v T±
— K - codt]

GC

K denoting the specific conducibility of the mass. On the other
hand, the surface co permits a quantity of heat to escape into the
air, during the time dt, equal to hvcodt ; h being the conducibility
relative to atmospheric air. Thus the flow of heat at the end of
the normal has two different expressions, that is to say :

hvcodt and — K - — codt ;

hence these two quantities are equal ; and it is by the expression
of this equality that the condition relative to the surface is in
troduced into the analysis.

147. We have

, £ . dv ^ dv ~ dv £
w — v + ov = v + -y- ox + -j- oy -f- -j~ oz.
ax dy dz

Now, it follows from the principles of geometry, that the co
ordinates $x, &/, &z, which fix the position of the point v of the
normal relative to the point ^ satisfy the following conditions :

We have therefore

w

1 / dv dv dv\ <*
-v = - (m-j- + n-j- + p^-) oz:
p\ dx dy * dz

SECT. VII.] GENERAL SURFACE EQUATION. 11?

we have also

,^-s — Bia &s2=-(m2

or a. = ^ &z , denoting by q the quantity (m2 + n* + p2)'" ,

w — vfdv dv , cfaA 1
hence - = [m -,— + n-j-+p-j- 1- ;

a \ dx dy L dzj q

consequently the equation

becomes the followin

dv dv

This equation is definite and applies only to points at the
surface ; it is that which must be added to the general equation of
the propagation of heat (A), and to the condition which deter
mines the initial state of the solid ; m, n, p, q, are known functions
of the co-ordinates of the points on the surface.

148. The equation (B) signifies in general that the decrease of
the temperature, in the direction of the normal, at the boundary of
the solid, is such that the quantity of heat which tends to escape
by virtue of the action of the molecules, is equivalent always to
that which the body must lose in the medium.

The mass of the solid might be imagined to be prolonged,
in such a manner that the surface, instead of being exposed to the
air, belonged at the same time to the body which it bounds, and
to the mass of a solid envelope which contained it. If, on this
hypothesis, any cause whatever regulated at every instant the
decrease of the temperatures in the solid envelope, and determined
it in such a manner that the condition expressed by the equation
(B) was always satisfied, the action of the envelope would take the

1 Let .ZV be the normal,

the rest as in the text. [B. L. E.]

dv m dv

-7T7 = — -T- + &c. ;

<LV q dx

118 THEORY OF HEAT. [CHAP. II.

place of that of the air, and the movement of heat would be the
same in either case : we can suppose then that this cause exists,
and determine on this hypothesis the variable state of the solid ;
which is what is done in the employment of the two equations
(A) and (B).

By this it is seen how the interruption of the mass and the
action of the medium, disturb the diffusion of heat by submitting
it to an accidental condition.

149. We may also consider the equation (B), which relates
to the state of the surface under another point of view : but we
must first derive a remarkable consequence from Theorem in.
(Art. 140). We retain the construction referred to in the corollary
of the same theorem (Art. 141). Let x, y, z be the co-ordinates
of the point p, and

x+Sx, y + %, z + §z

those of a point q infinitely near to p, and taken on the straight
line in question : if we denote by v and w the temperatures of the
two points p and q taken at the same instant, we have

, 5 , dv £ , dv 2 , dv 5,

w = v 4- bv = v + -j- ox + -j- o y + -y- oz ;
dx dy ' dz

hence the quotient

Sv dv 8x dv dy dv §z

-5- = -j- -Z- + -J- * + j- -F" i

be dx be dx ce dz ce

thus the quantity of heat which flows across the surface <y placed
at the point m, perpendicular to the straight line, is

dv Sx dv Sv dv Sz

7 r\

The first term is the product of — K-j~ by dt and by CD -K-.

dx 06

The latter quantity is, according to the principles of geometry, the
area of the projection of co on the plane of y and z ; thus the
product represents the quantity of heat which would flow across
the area of the projection, if it were placed at the point p perpen
dicular to the axis of x.

SECT. VII.] GENEKAL SURFACE EQUATION. 119

7 rs

The second term — K -r- co ~- dt represents the quantity of

heat which would cross the projection of a), made on the plane of
x and z, if this projection were placed parallel to itself at the
point p.

7 rj

Lastly, the third term - K -j- co -~-dt represents the quantity

of heat which would flow during the instant dt, across the projec
tion of o> on the plane of so and y, if this projection were placed at
the point p, perpendicular to the co-ordinate z.

By this it is seen that the quantity of heat which flows across
every infinitely small part of a surface drawn in the interior of the
solid, can always be decomposed into three other quantities of flow,
which penetrate the three orthogonal projections of the surface, along
the directions perpendicular to the planes of the projections. The
result gives rise to properties analogous to those which have
been noticed in the theory of forces.

150. The quantity of heat which flows across a plane surface
ft>, infinitely small, given in form and position, being equivalent
to that which would cross its three orthogonal projections, it fol
lows that, if in the interior of the solid an element be imagined of
any form whatever, the quantities of heat which pass into this
polyhedron by its different faces, compensate each other recipro
cally: or more exactly, the sum of the terms of the first order,
which enter into the expression of the quantities of heat received
by the molecule, is zero ; so that the heat which is in fact accumu
lated in it, and makes its temperature vary, cannot be expressed
except by terms infinitely smaller than those of the first order.

This result is distinctly seen when the general equation (A)
has been established, by considering the movement of heat in
a prismatic molecule (Articles 127 and 142) ; the demonstration
may be extended to a molecule of any form whatever, by sub
stituting for the heat received through each face, that which its
three projections would receive.

In other respects it is necessary that this should be so : for, if
one of the molecules of the solid acquired during each instant a
quantity of heat expressed by a term of the first order, the varia
tion of its temperature would be infinitely greater than that of

120

THEORY OF HEAT.

[CHAP. II.

other molecules, that is to say, during each infinitely small instant
its temperature would increase or decrease by a finite quantity,
which is contrary to experience.

151. We proceed to apply this remark to a molecule situated
at the external surface of the solid.

Fig. 6.

a

Through a point a (see fig. 6), taken on the plane of x and y,
draw two planes perpendicular, one to the axis of x the other to
the axis of y. Through a point b of the same plane, infinitely
near to a, draw two other planes parallel to the two preceding
planes ; the ordinates z, raised at the points a, b, c, d, up to the
external surface of the solid, will mark on this surface four points
a', b', c', d', and will be the edges of a truncated prism, whose base
is the rectangle abed. If through the point a which denotes the
least elevated of the four points a', b', c, dr, a plane be drawn
parallel to that of x and y, it will cut off from the truncated prism
a molecule, one of whose faces, that is to say ab'c'd', coincides
with the surface of the solid. The values of the four ordinates
a a', cc, dd'} bb' are the following :

aaf — z,

77' i j

bb = z -f- -y- dx -f- -j- d>/.
dx dy J

SECT. VII.] GENERAL SURFACE EQUATION. 121

152. One of the faces perpendicular to x is a triangle, and
the opposite face is a trapezium. The area of the triangle is

1 , ch

and the flow of heat in the direction perpendicular to this surface
-y-

CLOO

being — K -y- we have, omitting the factor dt,

dz

as the expression of the quantity of heat which in one instant
passes into the molecule, across the triangle in question.
The area of the opposite face is

1 j f dz , , dz , dz , \
- ay [ -j— ax + -y- ax + -j~ ay ,

2 9 \dx dx dy yj

CM ?7

and the flow perpendicular to this face is also — K-J-, suppress
ing terms of the second order infinitely smaller than those of the
first; subtracting the quantity of heat which escapes by the second
face from that which enters by the first we find

Trdv dz j j

K -7- -j- dx dy.
dx dx

This term expresses the quantity of heat the molecule receives
through the faces perpendicular to x.

It will be found, by a similar process, that the same molecule
receives, through the faces perpendicular to y, a quantity of heat

, , vr dv dz , ,
equal to K -^ j— dx dy.

The quantity of heat which the molecule receives through the

dv

rectangular base is — K-j-dx dy. Lastly, across the upper sur
face a'Vc'd', a certain quantity of heat is permitted to escape,
equal to the product of hv into the extent co of that surface.
The value of o> is, according to known principles, the same as that

of dx dy multiplied by the ratio - ; e denoting the length of the
normal between the external surface and the plane of x and ?/, and

fdz\* (dz

4- l-T- + (-—

j \dy

122 THEORY OF HEAT. [CHAP. II.

hence the molecule loses across its surface a'b'c'd' a quantity of
heat equal to hv dx dy - .

Now, the terms of the first order which enter into the expression
of the total quantity of heat acquired by the molecule, must cancel
each other, in order that the variation of temperature may not be
at each instant a finite quantity ; we must then have the equation

dz dv dz , , dv

j — j— ^ — j-

dx dx y dy dy

, , dv , , \ , e , ,
ax dy — r dx dy} — hv-dxdy = 0,
*\ d* * *J z

he dv dz dv dz dv
or -==,v - — -j- -j— + -j— -j --- j- .
K z dx dx dy dy dz

153. Substituting for -r- and -7- their values derived from
& dx dy

the equation

mdx 4- ndy -\-pdz = 0,

and denoting by q the quantity

(w'+w'+p8) ,
we have

dv dv dv

thus we know distinctly what is represented by each of the
terms of this equation.

Taking them all with contrary signs and multiplying them
by dx dy, the first expresses how much heat the molecule receives
through the two faces perpendicular to x, the second how much
it receives through its two faces perpendicular to y, the third
how much it receives through the face perpendicular to z, and
the fourth how much it receives from the medium. The equation
therefore expresses that the sum of all the terms of the first
order is zero, and that the heat acquired cannot be represented
except by terms of the second order.

154. To arrive at equation (B), we in fact consider one
of the molecules whose base is in the surface of the solid, as
a vessel which receives or loses heat through its different faces.
The equation signifies that all the terms of the first order which

SECT. VIII.] GENERAL EQUATIONS APPLIED. 123

enter into the expression of the heat acquired cancel each other ;
so that the gain of heat cannot be expressed except by terms
of the second order. We may give to the molecule the form,
either of a right prism whose axis is normal to the surface of the
solid, or that of a truncated prism, or any form whatever.

The general equation (A), (Art. 142) supposes that all the
terms of the first order cancel each other in the interior of the
mass, which is evident for prismatic molecules enclosed in the
solid. The equation (B), (Art. 147) expresses the same result
for molecules situated at the boundaries of bodies.

Such are the general points of view -from which we may look
at this part of the theory of heat.

„, ' dv K fd*v d*v <Fv\ ,,

The equation ^ = m (^ + jf+&) represents the move-

ment of heat in the interior of bodies. It enables us to ascer
tain the distribution from instant to instant in all substances
solid or liquid ; from it we may derive the equation which
belongs to each particular case.

In the two following articles we shall make this application
to the problem of the cylinder, and to that of the sphere.

SECTION VIII.

Application of the general equations.

155. Let us denote the variable radius of any cylindrical
envelope by r, and suppose, as formerly, in Article 118, that
all the molecules equally distant from the axis have at each
instant a common temperature ; v will be a function of r and t ;
r is a function of y, z, given by the equation r2 = yz + z*. It is
evident in the first place that the variation of v with respect

73

to x is nul : thus the term -j-s must be omitted. We shall have

dx*

then, according to the principles of the differential calculus, the
equations

dv_dvdr , d*v _ d?v_ (dr\* dv
Ty ~ dr Ty J ~df~dr* [dy) + d

dv dv dr , d2v

~r~ = i — r aud ~ra
dz dr dz dzz

d*v fdr\* dv fd*r\

= ~rr I ~5~ I + T~ I -i~« ;

dr* \dz) dr \dz*J '

124 THEORY OF HEAT. [CHAP. II.

whence

<Pv (Fv__d*v (fdr\* (dr\* .dvfd^r dfr
dy* + dz* dr2 \\cty) + \dz) + dr \dy* +

In the second member of the equation, the quantities

dr dr d*r d*r
Ty ' Tz ' ~dtf ' 2? J

must be replaced by their respective values ; for which purpose
we derive from the equation yz + z* = rz,

dr fdr\* d*r

y—T-r and 1=^-1 + r -j—» ,
dy \dyj dy* '

dr fdr\* d*r

z = r-j- and 1 = — +r-r-» ,
dz \dzj dz

and consequently

The first equation, whose first member is equal to r2, gives

the second gives, when we substitute for

fdr\* /AY
\dy) + (&)
its value 1,

If the values given by equations (b) and (c) be now substi
tuted in (a), we have

(Fv d?v dh Idv
dtf + dz*~drt + r dr'

Hence the equation which expresses the movement of heat
in the cylinder, is

dv_J?i(d^) ldv\
dt ~~ CD Ur2 * r dr) '
as was found formerly, Art. 119.

SECT. VIII.] EQUATIONS APPLIED TO A SPHERE. 125

We might also suppose that particles equally distant from
the centre have not received a common initial temperature ;
in this case we should arrive at a much more general equation.

156. To determine, by means of equation (A), the movement
of heat in a sphere which has been immersed in a liquid, we
shall regard v as a function of r and t ; r is a function of x, y, z,
given by the equation

r being the variable radius of an envelope. We have then

dv dv dr , dzv dzv fdr\z dv d*r
•j- — -y- -r- and -r-2 = -i-g ( -=- ) + -y- -=— ,
au; ar dx dx dr \dxj dr dx

dv dv dr dzv_dzv/dr\2 dv d~r

dv _ dv dr , d2v __ d*v /dr\2 dv d*r
~ a ~ +

Making these substitutions in the equation

dv_Jt_(d*v d*v <
dt~ CD(dx* + dyz +

we shall have

dv K <Pv (dr\* dr\* dz\* dv (d

The equation x* + y2 + z2 = r2 gives the following results ;
dr drz

dr . fdr\* tfr

y — r ~r~ and i = I -j- ) + T -;-=

z

j

d-y \dy]

dr fdr\z tfr

z — r-^~ and 1 = -^— + r -j-$ .
dz \dzj dzz

The three equations of the first order give :

126 THEORY OF HEAT. [CHAP. II.

The three equations of the second order give :

dr\*
dy *V ' dxz dy*

and substituting for

_(dr\ fdr\

" \dx) + \dy) + ' T z + * + ' ''

dx
its value 1, we have

ffr

Making these substitutions in the equation (a) we have the
equation

~dt^UD |<F + r ~<FJ'

which is the same as that of Art. 114.

The equation would contain a greater number of terms, if we
supposed molecules equally distant from the centre not to have
received the same initial temperature.

We might also deduce from the definite equation (B), the
equations which express the state of the surface in particular
cases, in which we suppose solids of given form to communicate
their heat to the atmospheric air ; but in most cases these equa
tions present themselves at once, and their form is very simple,
when the co-ordinates are suitably chosen.

SECTION IX.
General Remarks.

157. The investigation of the laws of movement of heat in
solids now consists in the integration of the equations which we
have constructed ; this is the object of the following chapters.
We conclude this chapter with general remarks on the nature
of the quantities which enter into our analysis.

In order to measure these quantities and express them nume
rically, they must be compared with different kinds of units, five

SECT. IX.] GENERAL REMARKS. 127

in number, namely, the unit of length, the unit of time, that of
temperature, that of weight, and finally the unit which serves to
measure quantities of heat. For the last unit, we might have
chosen the quantity of heat which raises a given volume of a
certain substance from the temperature 0 to the temperature 1.
The choice of this unit would have been preferable in many
respects to that of the quantity of heat required to convert a mass
of ice of a given weight, into an equal mass of water at 0, without
raising its temperature. We have adopted the last unit only
because it had been in a manner fixed beforehand in several works
on physics ; besides, this supposition would introduce no change
into the results of analysis.

158. The specific elements which in every body determine
the measurable effects of heat are three in number, namely, the
conducibility proper to the body, the conducibility relative to the
atmospheric air, and the capacity for heat. The numbers which
express these quantities are, like the specific gravity, so many
natural characters proper to different substances.

We have already remarked, Art. 36, that the conducibility of
the surface would be measured in a more exact manner, if we had
sufficient observations on the effects of radiant heat in spaces
deprived of air.

It may be seen, as has been mentioned in the first section of
Chapter L, Art. 11, that only three specific coefficients, K, h, C,
enter into the investigation ; they must be determined by obser
vation ; and we shall point out in the sequel the experiments
adapted to make them known with precision.

159. The number C which enters into the analysis, is always
multiplied by the density D, that is to say, by the number of
units of weight which are equivalent to the weight of unit of
volume ; thus the product CD may be replaced by the coeffi
cient c. In this case we must understand by the specific capacity
for heat, the quantity required to raise from temperature 0 to
temperature 1 unit of volume of a given substance, and not unit of
weight of that substance.

With the view of not departing from the common definition,
we have referred the capacity for heat to the weight and not to

128 THEORY OF HEAT. [CHAP. II.

the volume ; but it would be preferable to employ the coefficient c
which we have just denned ; magnitudes measured by the unit
of weight would not then enter into the analytical expressions :
we should have to consider only, 1st, the linear dimension x, the
temperature v, and the time t\ 2nd, the coefficients c, h, and K.
The three first quantities are undetermined, and the three others
are, for each substance, constant elements which experiment
determines. As to the unit of surface and the unit of volume,
they are not absolute, but depend on the unit of length.

160. It must now be remarked that every undetermined
magnitude or constant has one dimension proper to itself, and
that the terms of one and the same equation could not be com
pared, if they had not the same exponent of dimension. We have
introduced this consideration into the theory of heat, in order to
make our definitions more exact, and to serve to verify the
analysis; it is derived from primary notions on quantities; for
which reason, in geometry and mechanics, it is the equivalent
of the fundamental lemmas which the Greeks have left us with
out proof.

161. In the analytical theory of heat, every equation
expresses a necessary relation between the existing magnitudes
x, t, v, c, h, K. This relation depends in no respect on the choice
of the unit of length, which from its very nature is contingent,
that is to say, if we took a different unit to measure the linear
dimensions, the equation (E} would still be the same. Suppose
then the unit of length to be changed, and its second value to be
equal to the first divided by m. Any quantity whatever x which
in the equation (E) represents a certain line ab, and which, con
sequently, denotes a certain number of times the unit of length,
becomes inx, corresponding to the same length ab ; the value t
of the time, and the value v of the temperature will not be
changed ; the same is not the case with the specific elements

h, K, c\ the first, h, becomes — , ; for it expresses the quantity of

i(Ylt

heat which escapes, during the unit of time, from the unit of sur
face at the temperature 1. If we examine attentively the nature
of the coefficient K, as we have defined it in Articles 68 and 135,

SECT. IX.] UNITS AND DIMENSIONS. 129

TS-

we perceive that it becomes — : for the flow of heat varies

m

directly as the area of the surface, and inversely as the distance
between two infinite planes (Art. 72). As to the coefficient c

which represents the product CD, it also depends on the unit of

£
length and becomes — 3; hence equation (E) must undergo no

change when we write mx instead of x, and at the same time

- , — = , — 3 , instead of K, h, c - the number m disappears after
m m~ m

these substitutions : thus the dimension of x with respect to the
unit of length is 1, that of K is — 1, that of h is — 2, and that of c
is — .3. If we attribute to each quantity its own exponent of di
mension, the equation will be homogeneous, since every term will
have the same total exponent. Numbers such as $, which repre
sent surfaces or solids, are of two dimensions in the first case,
and of three dimensions in the second. Angles, sines, and other
trigonometrical functions, logarithms or exponents of powers, are,
according to the principles of analysis, absolute numbers which do
not change with the unit of length ; their dimensions must there
fore be taken equal to 0, which is the dimension of all abstract
numbers.

If the unit of time, which was at first 1, becomes -, the number

n

t will become nt, and the numbers x and v will not change. The

coefficients K, h, c will become — , - , c. Thus the dimensions

n n

of x, t, v with respect to the unit of time are 0, 1, 0, and those of
Kt h, c are - 1, - 1, 0.

If the unit of temperature be changed, so that the temperature
1 becomes that which corresponds to an effect other than the
boiling of water ; and if that effect requires a less temperature,
which is to that of boiling water in the ratio of 1 to the number p-
v will become vp, x and t will keep their values, and the coeffi
cients K. h, c will become — , - . - .

P P P

The following table indicates the dimensions of the three

undetermined quantities and the three constants, with respect
to each kind of unit.

F. H. 9

130

THEORY OF HEAT.

[CH. II. SECT. IX.

Quantity or Constant.

Length.

Duration.

Temperature.

Exponent of dimension of x ...

1

0

0

>> » * •••

0

1

0

„ ,, v ...

0

0

1

The specific conducibility, K ...

-1

-1

-1

The surface conducibility, h ...

-2

-1

-1

The capacity for heat, c ...

-3

0

1

162. If we retained the coefficients C and D, whose product
has been represented by c, we should have to consider the unit of
weight, and we should find that the exponent of dimension, with
respect to the unit of length, is — 3 for the density D, and 0
for G.

On applying the preceding rule to the different equations and
their transformations, it will be found that they are homogeneous
with respect to each kind of unit, and that the dimension of every
angular or exponential quantity is nothing. If this were not the
case, some error must have been committed in the analysis, or
abridged expressions must have been introduced.

If, for example, we take equation (6) of Art. 105,

dv _ K d*v hi
dt ~~GD ~da?~ CDSV'

we find that, with respect to the unit of length, the dimension of
each of the three terms is 0 ; it is 1 for the unit of temperature,
and — 1 for the unit of time.

/ 2/2»

In the equation v = Ae~x & of Art. 76, the linear dimen
sion of each term is 0, and it is evident that the dimension of the

exponent x A/ ^~ is always nothing, whatever be the units of
length, time, or temperature.

CHAPTER III.

•

PROPAGATION OF HEAT IN AN INFINITE RECTANGULAR SOLID.

SECTION I.

Statement of the problem.

163. PROBLEMS relative to the uniform propagation, or to
the varied movement of heat in the interior of solids, are reduced,
by the foregoing methods, to problems of pure analysis, and
the progress of this part of physics will depend in consequence
upon the advance which may be made in the art of analysis.
The differential equations which we have proved contain the
chief results of the theory ; they express, in the most general
and most concise manner, the necessary relations of numerical
analysis to a very extensive class of phenomena; and they
connect for ever with mathematical science one of the most
important branches of natural philosophy.

It remains now to discover the proper treatment of these
equations in order to derive their complete solutions and an
easy application of them. The following problem offers the
first example of analysis which leads to such solutions ; it
appeared to us better adapted than any other to indicate the
elements of the method which we have followed.

164. Suppose a homogeneous solid mass to be contained
between two planes B and G vertical, parallel, and infinite, and
to be divided into two parts by a plane A perpendicular to the
other two (fig. 7) ; we proceed to consider the temperatures of
the mass BAC bounded by the three infinite planes At B, C.
The other part B'AC' of the infinite solid is supposed to be a
constant source of heat, that is to say, all its points are main
tained at the temperature 1, which cannot alter. The two

9—2

132

THEORY OF HEAT.

[CHAP. III.

lateral solids bounded, one by the plane C and the plane A
produced, the other by the plane B and the plane A pro-

JB

-

y~ *
so

C

A

^
i

\c'

duced, have at all points the constant temperature 0, some
external cause maintaining them always at that temperature;
lastly, the molecules of the solid bounded by A, B and C have
the initial temperature 0. Heat will pass continually from the
source A into the solid BAG, and will be propagated there in
the longitudinal direction, which is infinite, and at the same
time will turn towards the cool masses B and C, which will ab
sorb great part of it. The temperatures of the solid BAG will
be raised gradually : but will not be able to surpass nor even
to attain a maximum of temperature, which is different for
different points of the mass. It is required to determine the
final and constant state to which the variable state continually
approaches.

If this final state were known, and were then formed, it would
subsist of itself, and this is the property which distinguishes
it from all other states. Thus the actual problem consists in
determining the permanent temperatures of an infinite rect
angular solid, bounded by two masses of ice B and G, and a
mass of boiling water A ; the consideration of such simple and
primary problems is one of the surest modes of discovering the
laws of natural phenomena, and we see, by the history of the
sciences, that every theory has been formed in this manner.

165. To express more briefly the same problem, suppose
a rectangular plate BA C, of infinite length, to be heated at its
base A, and to preserve at all points of the base a constant

SECT. I.] INFINITE RECTANGULAR SOLID. 133

temperature 1, whilst each of the two infinite sides B and C,
perpendicular to the base A, is submitted also at every point
to a constant temperature 0; it is required to determine what
must be the stationary temperature at any point of the plate.

It is supposed that there is no loss of heat at the surface
of the plate, or, which is the same thing, we consider a solid
formed by superposing an infinite number of plates similar to
the preceding : the straight line Ax which divides the plate
into two equal parts is taken as the axis of x, and the co-ordinates
of any point m are x and y ; lastly, the width A of the plate
is represented by 21, or, to abridge the calculation, by IT, the
value of the ratio of the diameter to the circumference of a
circle.

Imagine a point m of the solid plate B A (7, whose co-ordinates
are x and y, to have the actual temperature v, and that the
quantities v, which correspond to different points, are such that
110 change can happen in the temperatures, provided that the
temperature of every point of the base A is always 1, and that
the sides B and C retain at all their points the temperature 0.

If at each point m a vertical co-ordinate be raised, equal to
the temperature v, a curved surface would be formed which
would extend above the plate and be prolonged to infinity.
We shall endeavour to find the nature of this surface, which
passes through a line drawn above the axis of y at a distance
equal to unity, and which cuts the horizontal plane of xy along
two infinite straight lines parallel to x.

166. To apply the general equation

di CD \dx2 dy2 d
we must consider that, in the case in question, abstraction is

72

made of the co-ordinate z, so that the term -y-n must be omitted ;

az

with respect to the first member -=- , it vanishes, since we wish to
determine the stationary temperatures ; thus the equation which

134 THEORY OF HEAT. [CHAP. III.

belongs to the actual problem, and determines the properties
of the required curved surface, is the following :

The function of a? and y> <f> (x, y), which represents the per
manent state of the solid BA G, must, 1st, satisfy the equation
(a) ; 2nd, become nothing when we substitute — J TT or + \ir for y,
whatever the value of x may be ; 3rd, must be equal to unity
when we suppose x = 0 and y to have any value included between
— J TT and + i TT.

Further, this function <£ (x, y) ought to become extremely
small when we give to x a very large value, since all the heat
proceeds from the source A.

167. In order to consider the problem in its elements, we
shall in the first place seek for the simplest functions of x
and y, which satisfy equation (a) ; we shaTT then generalise the
value of v in order to satisfy all the stated conditions. By this
method the solution will receive all possible extension, and we
shall prove that the problem proposed admits of no other
solution.

Functions of two variables often reduce to less complex ex
pressions, when we attribute to one of the variables or to both
of them infinite values ; this is what may be remarked in alge
braic functions which, in this particular case, take the form of
the product of a function of x by a function of y.

We shall examine first if the value of v can be represented
by such a product ; for the function v must represent the state
of the plate throughout its whole extent, and consequently that
of the points whose co-ordinate x is infinite. We shall then
write v = F(x)f(y}\ substituting in equation (a) and denoting

by F" (x) and by/' (y\ we shall have

(*) ,/'(y)_0.

'

we then suppose „ \^ = m and r^ = — m>> m being any

SECT. I.] INFINITE RECTANGULAR PLATE. 135

constant quantity, and as it is proposed only to find a particular
value of v, we deduce from the preceding equations F(x) = e~mx}
/(?/)= cos my.

168. We could not suppose m to be a negative number,
and we must necessarily exclude all particular values of v, into
which terms such as emx might enter, m being a positive number,
since the temperature v cannot become infinite when x is in
finitely great. In fact, no heat being supplied except from the
constant source Ay only an extremely small portion can arrive
at those parts of space which are very far removed from the
source. The remainder is diverted more and more towards the
infinite edges B and C, and is lost in the cold masses which
bound them.

The exponent m which enters into the function e~"lrcosmy
is unknown, and we may choose for this exponent any positive
number: but, in order that v may become nul on making
y = — | TT or y = + |- TT, whatever x may be, m must be taken
to be one of the terms of the series, 1, 3, 5, 7, &c. ; by this
means the second condition will be fulfilled.

169. A more general value of v is easily formed by adding
together several terms similar to the preceding, and we have

le~3x cos 3j/ -f- ce~5x cos 5y + de~lx cos 7y + &c. . . f. . .

It is evident that the function v denoted by $ (x, y) satis!
the equation -^ + -=-£ = 0, and the condition <f> (x, ± J TT) = 0.

A third condition remains to be fulfilled, which is expressed thus,
<f> (0, y) = 1, and it is essential to remark that this result must
exist when we give to y any value whatever included between
— \ TT and -f J TT. Nothing can be inferred as to the values
which the function <f> (0, y) would take, if we substituted in place
of y a quantity not included between the limits — J TT and -f J TT.
Equation (b) must therefore be subject to the following condition :

1 = a cos y + b cos 3^ + c cos 5y + d cos 7y + &c.

The coefficients, a, b, c, d, &c., whose number is infinite, are
determined by means of this equation.

The second member is a function of y, which is equal to 1

136 THEOEY OF HEAT. [CHAP. III.

so long as the variable y is included between the limits — \ TT
and + ^ TT. It may be doubted whether such a function exists,
but this difficulty will be fully cleared up by the sequel.

170. Before giving the calculation of the coefficients, we
may notice the effect represented by each one of the terms of
the series in equation (b).

Suppose the fixed temperature of the base A^ instead of
being equal to unity at every point, to diminish as the point
of the line A becomes more remote from the middle point,
being proportional to the cosine of that distance ; in this case
it will easily be seen what is the nature of the curved surface,
whose vertical ordinate expresses the temperature v or fy (x, ?/).
If this surface be cut at the origin by a plane perpendicular
to the axis of x, the curve which bounds the section will have
for its equation v = a cos y ; the values of the coefficients will
be the following :

a = a, Z>=0, c = 0, d= 0,

and so on, and the equation of the curved surface will be

v = ae~x cos y.

If this surface be cut at right angles to the axis of y, the
section will be a logarithmic spiral whose convexity is turned
towards the axis; if it be cut at right angles to the axis of x,
the section will be a trigonometric curve whose concavity is
turned towards the axis.

It follows from this that the function -7-5- is always positive,

ctx

d*v
and -^-3 is always negative. Now • the quantity of heat which

a molecule acquires in consequence of its position . between two

others in the direction of x is proportional to the value of -^

ctoc

(Art. 123) : it follows then that the intermediate molecule receives
from that which precedes it, in the direction of x, more heat than
it communicates to that which follows it. But, if the same mole
cule be considered as situated between two others in the direction

of y, the function --a being negative, it appears that the in-

SECT. II.] TRIGONOMETRIC SERIES. 1 37

termediate molecule communicates to that which follows it more
heat than it receives from that which precedes it. Thus it
follows that the excess of the heat which it acquires in the direc
tion of x, is exactly compensated by that whicn" it loses in the

direction of ?/. as the equation -^-2 + -y-2=0 denotes. Thus

ax dy

then the route followed by the heat which escapes from the
source A becomes known. It is propagated in the direction
of x, and at the same time it is decomposed into two parts,
one of which is directed towards one of the edges, whilst the
other part continues to separate from the origin, to be decomposed
like the preceding, and so on to infinity. The surface which
we are considering is generated by the trigonometric curve which
corresponds to the base A, moved with its plane at right angles to
the axis of x along that axis, each one of its ordinates de
creasing indefinitely in proportion to successive powers of the
same fraction.

Analogous inferences might be drawn, if the fixed tempera
tures of the base A were expressed by the term

b cos 3y or c cos 5y, &c. ;

and in this manner an exact idea might be formed of the move
ment of heat in the most general case ; for it will be seen by
the sequel that the movement is always compounded of a multi
tude of elementary movements, each of which is accomplished
as if it alone existed.

SECTION II.

First example of the use of trigonometric series in the theory

of heat.

171. Take now the equation

1 = a cos y + b cos oy + c cos oy + d cos 7y + &c.,

in which the coefficients a, b, c, d, &c. are to be determined.
In order that this equation may exist, the constants must neces-

138 THEORY OF HEAT. [CHAP. III.

sarily satisfy the equations which are obtained by successive

differentiations ; whence the following results,

1 = a cos y + b cos 3y + c cos 5y + d cos 1y -f &c.,
0 = a sin y + 3b sin 3y + 5c sin 5y + 7d sin 7y + &c.,
0 = a cos y + 32& cos 3# + 52c cos 5^ + 72c£ cos 7?/ + &c.,

0 = a sin y + 336 sin 3y + 53c sin oy + Td sin 7y + &c.,
and so on to infinity.

These equations necessarily hold when y = 0, thus we have

1 = a+ 5+ c+ cl+ e+ f+ 0+...&C.,
0 = a + 32t> + 52c + 72d^ + 92e + H2/+ ... &c.,

0 = a + 345 + 54c + 74J+946+ ... &c.,

0 = a + 366 + 5Gc + 76^+ ... &c.,

0 = a + 386 + 58c -f . . . «fec.,

&c.

The number of these equations is infinite like that of the
unknowns a, b, c, d, e, ... &c. The problem consists in eliminating
all the unknowns, except one only.

172. In order to form a distinct idea of the result of these
eliminations, the number of the unknowns a, b, c, d, ...&c., will
be supposed at first definite and equal to m. We shall employ
the first m equations only, suppressing all the terms containing
the unknowns which follow the m first. If in succession m
be made equal to 2, 3, 4, 5, and so on, the values of the un
knowns will be found on each one of these hypotheses. The
quantity a, for example, will receive one value for the case
of two unknowns, others for the cases of three, four, or successively
a greater number of unknowns. It will be the same with the
unknown 6, which will receive as many different values as there
have been cases of elimination ; each one of the other unknowns
is in like manner susceptible of an infinity of different values.
Now the value of one of the unknowns, for the case in which
their number is infinite, is the limit towards which the values
which it receives by means of the successive eliminations tend.
It is required then to examine whether, according as the number
of unknowns increases, the value of each one of a, b, c, d ... &c.
does not converge to a finite limit which it continually ap
proaches.

SECT. II.] DETERMINATION "OF COEFFICIENTS. 139

Suppose the six following equations to be employed :
1 = a + b + c + d + e + f + &c.,
0 = a + 32Z> + 52c +Td +92e +H2/+&c.,
0 = a + 34& + 54c + Td + 94e + ll4/ + &c.,
0 = a + 366 + 56c + Td + 96e + ll6/ -I- &c.,
0 = a + 38£ -f 58c + 78d + 98e + ll8/ + &c ,
0 = a + 3106 + 510c + 7wd + 910e + ll10/ + &c.

The five equations which do not contain /are :
Il2=a(ll2-l2)+ Z>(H2-32)+ c(H2-52)+ J(ll2-72)+ e(H2-92);

0=a(ll2-l2)+366(ir-32)+56c(ll2-52)+76cZ(ll2-72)+96e(ll2-92),
0=a(ll2-r)+386(ir-32)+58c(ll2-52)+78^(ir-72)+9^(ll2-92).

Continuing the elimination we shall obtain the final equation
in a, which is :

a (ll2- 12) (92- 12) (72- 12) (52 - 12) (32 - I2) = ll2. 92. 72. 52. 32. 12.

173. If we had employed a number of equations greater
by unity, we should have found, to determine a, an equation
analogous to the preceding, having in the first member one
factor more, namely, 132 — I2, and in the second member 132
for the new factor. The law to which these different values of
a are subject is evident, and it follows that the value of a which
corresponds to an infinite number of equations is expressed thus :
32 52 72 92 ,

/Vrp

•

_ 3 . 3 5.57.7 9.9 11 .11
~ 2T4 ' 476 ' 6T8 ' 8710 ' 10TT2 '

Now the last expression is known and, in accordance with
"Wallis* Theorem, wre conclude that a — — . It is required then
only to ascertain the values of the other unknowns.

174. The five equations which remain after the elimination
of / may be compared with the five simpler equations which
would have been employed if there had been only five unknowns.

140 THEORY OF HEAT. [CHAP. III.

The last equations differ from the equations of Art. 172, in
that in them e, d, c, b, a are found to be multiplied respec
tively by the factors

n2 -92 n'-jT2 ir- 52 ir-32 ir-r
"'•iv * ~iY~' n1 ' ~~Tr~' ir •

It follows from this that if we had solved the five linear
equations which must have been employed in the case of five
unknowns, and had calculated the value of each unknown, it
would have been easy to derive from them the value of the
unknowns of the same name corresponding to the case in which
six equations should have been employed. It would suffice to
multiply the values of e, d, c, b, a, found in the first case, by the
known factors. It will be easy in general to pass from the value
of one of these quantities, taken on the supposition of a certain
number of equations and unknowns, to the value of the same
quantity, taken in the case in which there should have been
one unknown and one equation more. For example, if the value
of e, found on the hypothesis of five equations and five unknowns,
is represented by E, that of the same quantity, taken in the case

II2

of one unknown more, will be E—- — —2. The same value,

j. JL — y

taken in the case of seven unknowns, will be, for the same reason,

11* -9* '13' -9"

and in the case of eight unknowns it will be
II2 132 152

E

11*— 9* '13* -9* "15* -9"

and so on. In the same manner it will suffice to know the
value of b, corresponding to the case of two unknowns, to derive
from it that of the same letter which corresponds to the cases
of three, four, five unknowns, &c. We shall only have to multiply
this first value of b by

52 72 92

.. &c.

52-32*72-32'9a-32

SECT. II.] DETERMINATION OF COEFFICIENTS. 141

Similarly if we knew the value of c for the case of three
unknowns, we should multiply this value by the successive factors

_r_ 9* ir

7*-52>92-52'ir-52"

We should calculate the value of d for the case of four unknowns
only, and multiply this value by

92 II2 132

The calculation of the value of a is subject to the same rule,
for if its value be taken for the case of one unknown, and multi
plied successively by

32 52 T 92

3* -1s" 5^T2' r^V' 9^T2'

the final value of this quantity will be found.

175. The problem is therefore reduced to determining the
value of a in the case of one unknown, the value of b in the case
of two unknowns, that of c in the case of three unknowns, and so
on for the other unknowns.

It is easy to conclude, by inspection only of the equations and
without any calculation, that the results of these successive elimi
nations must be

rt =

1,

I2

I.

o —

!2-32'

I2

32

c —

r-52 •

32 - 52 '

i2

32 52

d

r-T *

32 - T ' 52 - 72 '

i2

32 52 72

e —

I2 - 92 '

02 Q2 ' £* (JH ' IT* Q2

176. It remains only to multiply the preceding quantities by
the series of products which ought to complete them, and which
we have given (Art. 174). We shall have consequently, for the

142 THEORY OF HEAT. [CHAP. III.

final values of the unknowns a, b, c, d, e, f, &c., the following
expressions :

a —
7,

I2

52

72

92

II2

u —

r-3a *

52-32 '

72.32 •

92 _. 32 •

H2-32

C —

i2

32

72

92

112 &c

I2 - 52 •

I2

32-52 '
32

52

92 - 52 '
92

ir-52 "
n2

l» - 72 '

32-72 '

52-72 '

9* -7* '

H2-72 '

I2

32

52

72

II2 132

!2-92 '

32 - 92 '

52 - 92 '

72-92 '

1P-92 ' 132-92 '

I2

32

52

72

92 132

* •

i2-ir '

32-ll2

' 52-ll2

* 7*-ll

* ' 92-lla ' 132-112<

.

or,

n 4-1

S

1.3

5

5

7.7

&

r*

' 2

1.4 '

4.

6 '

6.8

1

.1

5.

5

7.

7

9

.9

2

. 4

' 2.

8 '

4.

10 '

6

.12

**J

1

.1

3.

3

7.

7

9

.9

11.

11 &c

1 4

.6

* 2.

8 *

2.

12 '

4

.14 '

6.

16 • '

*. l

.1

3.

3

5

.5

9.9

11

.11

_ _ _ ___

6.8 ' 4.10 ' 2.12 ' 2.16 ' 4.18

1.1 3.3 5.5 7.7 11.11 13.13
f 8 . 10 ' 6 . 12 ' 4 . 14 ' 2 . 16 ' 2 . 20 * 4 . 22

1.1 3.3 5.5 7.7 9.9 1313 15.15

10 . 12 ' 8 . 14 ' 6 . 16 ' 4 . 18 ' 2 . 20 ' 2 . 24 * 4 . 26

The quantity ^TT or a quarter of the circumference is equiva
lent, according to Wallis' Theorem, to

2.2 4.4 6.6 8.8 10.10 12.12 14.14
1 . 3 ' 3 . 5 ' 577 ' 77 9 ' ~97TT ' 11713 ' 137T5

SECT. II.

VALUES OF THE COEFFICIENTS.

If now in the values of a, b, c, d, &c., we notice what are the
factors which must be joined on to numerators and denominators
to complete the double series of odd and even numbers, we find
that the factors to be supplied are :

for 6
for c

for e

f^T* /

3.3

6 '
5 . 5

10 '
7.7

9.9
"18" '
11.11

-=V- , y whence we conclude .

a —

2

2

j

7T

b =

— 2 .

2

STT'

2

c =

2

5-Tr'

— 2

2

~

77T'

e =

2 .

2
977-'

/=

- 2.

2
UTT'

177. Thus the eliminations have been completely effected,
and the coefficients a, b} c, d, &c., determined in the equation

1 = a cos y + b cos 3?/ + c cos 5y + d cos 7y + e cos 9# + &c.

The substitution of these coefficients gives the following equa
tion :

7T

1

- = COS 7/ — - COS

1 c 1 K 1

-f ^COS 5?/— ^COS /^/+7^ COS

o / 9

- &c.

The second member is a function of y, which does not change
in value when we give to the variable y a value included between
— ^TT and -f |TT. It would be easy to prove that this series is
always convergent, that is to say that writing instead of y any
number whatever, and following the calculation of the coefficients,
we approach more and more to a fixed value, so that the difference
of this value from the sum of the calculated terms becomes less
than any assignable magnitude. Without stopping for a proof,

1 It is a little better to deduce the value of & in or, of c in &, &c. [E. L. E.]

2 The coefficients a, b, c, &c., might be determined, according to the methods
of Section vi. , by multiplying both sides of the first equation by cos y, cos 3?/,

cos 5v, &c., respectively, and integrating from --Trto +^TT, as was done by

& &

D. F. Gregory, Cambridge Mathematical Journal, Vol. i. p. 106. [A. F.]

144 THEORY OF HEAT. [CHAP. III.

which the reader may supply, we remark that the fixed value
which is continually approached is JTT, if the value attributed
to y is included between 0 and J-TT, but that it is — Jvr, if y is
included between \TT and |TT ; for, in this second interval, each
term of the series changes in sign. In general the limit of the
series is alternately positive and negative ; in other respects, the
convergence is not sufficiently rapid to produce an easy approxima
tion, but it suffices for the truth of the equation.

178. The equation

— ,3 cos ox + - cos ox — * cos 7% + &c.

O O /

belongs to a line which, having x for abscissa and y for ordinate, is
composed of separated straight lines, each of which is parallel to
the axis, and equal to the circumference. These parallels are
situated alternately above and below the axis, at the distance JTT,
and joined by perpendiculars which themselves make part of the
line. To form an exact idea of the nature of this line, it must be
supposed that the number of terms of the function

cos x — 7. cos 3x + - cos 5x — &c.
3 5

has first a definite value. In the latter case the equation

y = cos x — - cos 3x + - cos ox — &c.
o 5

belongs to a curved line which passes alternately above and below
the axis, cutting it every time that the abscissa x becomes equal
to one of the quantities

185

0, ± g 7T, + 2 7T, ± g 7T, &C.

According as the number of terms of the equation increases, the
curve in question tends more and more to coincidence with the
preceding line, composed of parallel straight lines and of perpen
dicular lines ; so that this line is the limit of the different curves
which would be obtained by increasing successively the number of
terms.

SECT. III.] REMARKS OX THE SERIES. 145

SECTION III.
Remarks on these series.

179. We may look at the same equations from another point
of view, and prove directly the equation

7 = cos x — - cos 3.r 4- - cos o.x — ^ cos 7x + Q cos 9# — &c.
The case where x is nothing is verified by Leibnitz' series,

7T 1 11 11 „

7 =1 - 7, ; + ^ - T= + 7: - &C.
4 3 o / 9

We shall next assume that the number of terms of the series

cos x — ^ cos 3# + -^ cos 5o: — ^ cos fa + &c.
o o /

instead of being infinite is finite and equal to m. We shall con
sider the value of the finite series to be a function of x and m.
We shall express this function by a series arranged according to
negative powers of m; and it will be found that the value of
the function approaches more nearly to being constant and inde
pendent of x, as the number m becomes greater.

Let y be the function required, which is given by the equation

y = cosx- Q cos 3.£ + - cos ox-^ cos 7x+...-- -cos (2wi — l)a?,

o o / Jim — 1

7?i, the number of terms, being supposed even. This equation
differentiated with respect to x gives

— -r- = sin x — sin 3# + sin ox — sin 7x + ...

+ sin (2??i — 3) x — sin (2wi — 1) x ;
multiplying by 2 sin Zx, we have

— 2 -y- sin 2# = 2 sin # sin 2# — 2 sin 3j? sin 2# + 2 sin 5# sin 2^ ...
cfo

+ 2 sin (2m - 3) or sin 2,z - 2 sin (2w - 1) x sin 2#.

F. H. 10

146 THEORY OF HEAT. , [CHAP. III.

Each term of the second member being replaced by the
difference of two cosines, we conclude that

- 2 -& sin 2# = cos (- a?) - cos 3#

— cos x + cos 5x
-}- cos 3# - cos 7x

— cos 5# + cos 9x

-f cos (2»i — 5) a? - cos (2w — 1) x
— cos (2m — 3x) -f cos (2m -f 1) #.
The second member reduces to

cos (2m + 1) x— cos (2m — 1) a-, or — 2 sin 2marsiu .r ;

1 */ sin %

hence

180. We shall integrate the second member by parts, dis
tinguishing in the integral between the factor smZmxdx which

must be integrated successively, and the factor or sec x

COSX

which must be differentiated successively ; denoting the results
of these differentiations by sec' x, sec" x, sec'" x, ... &c., we shall
have

1 1

2y = const. — ^-- cos 2?H# sec x + -— — :, sin 2mx sec x
2.m '2m

I
4- o»~* cos 2m# sec' x -f i\>c. ;

thus the value of y or

cos x — ;r cos 3x + - cos 5x — ^ cos 7x + . . . — cos (2m — 1 ) .r,

3 o 7 '2m - 1 ;

which is a function of x and m, becomes expressed by an infinite
series ; and it is evident that the more the number m increases,
the more the value of y tends to become constant. For this
reason, when the number m is infinite, the function y has a
definite value which is always the same, whatever be the positive

SECT. III.] PARTICULAR CASES. 1-V7

value of r, less than J-TT. Now, if the arc x be supposed nothing,
we have

1111

which is equal to JTT. Hence generally we shall have
1 111

- - 7T = COS X — ^ COS 3x + - COS OX — = COS

4 3 o 7

181. If in this equation we assume x = ~ _• , we find

-^L_-1 1_i_1 1 JL A J: £•
~ 3~5"7+9 + lI 13 15 ^C';

by giving to the arc x other particular values, we should find
other series, which it is useless to set down, several of which
have been already published in the works of Euler. If we
multiply equation (ft) by dx, and integrate it, we have

7TX . l-o 1 • - r 1 • * . fl

-T- = sm x — ^ sin 3^ + ^ sm °^ — T^> sm tx + &c.
4* o o 7"

Making in the last equation x = | TT, we find

a series already known. Particular cases might be enumerated
to infinity ; but it agrees better with the object of this work
to determine, by following the same process, the values of the
different series formed of the sines or cosines of multiple arcs.

182. Let
y = sin x - ^ sin 2x + ^ sin 3# - 7 sin 4# . . .

1 1

-i -- - sin [m — 1) x -- sin mr,
m — 1 7?i

m being any even number. We derive from this equation

-j- = cos x — cos 2# + cos ox — cos 4fx . . . + cos (m — 1) x — cos mx ;

10—2

148 THEORY OF HEAT. [CHAP. III.

multiplying by 2 sin x, and replacing each term of the second
member by the difference of two sines, we shall have

2 sin x -T- = sin (x + x)— sin (x - x)

- sin (2a? + x) + sin (2x - a;)
+ sin (3# + a?) — sin (3a? — x)

+ sin {(m — 1) a? - a;} — sin {(??? -f 1) a? — •#}
- sin (m.r + #) -f sin (ma? - x) ;
and, on reduction,

2 sin a? --,- = sin x + sin w# — sin (ma? + x} :
dx

the quantity sin mx - sin (?na; + a?),

or sin (wa? + J a? - Ja;) - sin (ma? -f 4# + iar),

is equal to - 2 sin \x cos (wia; + Ja;) ;

we have therefore

dn 1 sinA-a?
2

cos mx

sin a?

dy _ 1 cos (mx 4- i#) .
whence we conclude

or <to 2 2 cos

1 f cos (mx -}-fa)
] 2 cos fa

If we integrate this by parts, distinguishing between the

factor r- or sec \x, which must be successively differentiated,

cos^x

and the factor cos(mx+fa], which is to be integrated several
times in succession, we shall form a series in which the powers

of m + ^ enter into the denominators. As to the constant it
is nothing; since the value of y begins with that of x.

SECT. III.] SPECIAL SERIES. 149

It follows from this that the value of the finite series
sin x — g sin 2# + ^ sin 3x — - sin 5x -f- p sin 7x — . . . -- sin mx

differs very little from that of \xy when the number of terms
is very great ; and if this number is infinite, we have th,e known
equation

^ x — sin x — ^ sin 2x + ^ sin 3x - 7 sin 4# -f -? sin 5# — &c.

Zi £ o 4< o

From the last series, that which has been given above for
the value of JTT might also be derived.

183. Let now
y = ^ cos 2x — ^ cos 4x + - cos 6x - . . .

COS ~m -~®- COS ~"tx'

2m -2

Differentiating, multiplying by 2 sin 2x} substituting the
differences of cosines, and reducing, we shall have

ax cos x

f, r, i

or

r j j r^ sm (2??i +

= c - \dx tan x + \dx 2

J J cosx

integrating by parts the last term of the second member, and

supposing

equation

y

we suppose x nothing, we find

supposing m infinite, we have y = c + log cos x. If in the

y = ^ cos 2x - -r cos ±x + - cos Qx-- cos So; + . . . &c.

Z T) 0 o

therefore y = - log 2 + 5 log cos ir.

Thus we meet with the series given by Euler,

log (2 cos £#) = cos x - - cos 2# -f ^ cos 3x - -j cos 4^ + &c.

150 THEORY OF HEAT. [CHAP. III.

184. Applying the same process to tlie equation

y = sin #4- - sin 2x 4- ~ sin 5x 4 - sin 7x 4 &c.,
O D i

we find the following series, which has not been noticed,

-- TT = sin x 4 ^ sin ox 4 - sin ox 4 = sin 7x + -,- sin 9.£ 4- &c. l
4 3 o 7

It must be observed with respect to all these series, that
the equations which are formed by them do not hold except
when the variable x is included between certain limits. Thus
the function

cos x — -^ cos %x 4 v cos 5x — ^ cos 7x + &c.
3 o i

is not equal to JTT, except when the variable x is contained
between the limits which we have assigned. It is the same
with the series

sin x — - sin 2x 4- 0 sin %x — -r sin 4# 4 - sin ox — &c.
23 4 o

This infinite series, which is always convergent, has the value
\x so long as the arc x is greater than 0 and less than TT. But
it is not equal to %x, if the arc exceeds TT; it has on the contrary
values very different from \x ; for it is evident that in the in
terval from x — TT to x = 2ir, the function takes with the contrary
sign all the values which it had in the preceding interval from
x = 0 to x = TT. This series has been known for a long time,
but the analysis which served to discover it did not indicate
why the result ceases to hold when the variable exceeds TT.

The method which we are about to employ must therefore
be examined attentively, and the origin of the limitation to which
each of the trigonometrical series is subject must be sought.

185. To arrive at it, it is sufficient to consider that the
values expressed by infinite series are not known with exact
certainty except in the case where the limits of the sum of the
terms which complete them can be assigned ; it must therefore
be supposed that we employ only the first terms of these series,

1 This may be derived by integration from 0 to ir as in Art. 222. [R. L. E.]

SECT. III.] LIMITS OF THE REMAINDER. 151

and the limits between which the remainder is included must
be found.

We will apply this remark to the equation

1 1 1

y = cos x — - cos 3x + - cos ox — ^ cos tx ...
3 o 7

~

2m - 3 2m - 1

The number of terms is even and is represented by m ; from it

Zdy sin Zmx , . „ ,,

is derived the equation — — = - — , whence we may infer the

CtJO COS 00

value of y, by integration by parts. Now the integral fuvdx

may be resolved into a series composed of as many terms as

may be desired, u and v being functions of x. We may write, for
example,

I uvdx = c -f u I vdx — =- \dx Ivdx + -j— ., Idx I dxlvdx
J J dxj j dx'J J J

an equation which is verified by differentiation.

Denoting sin 2mx by v and sec x by u, it will be found that

2// = c — -T — sec x cos 2mx +^r-9 SQC'X sin 2??^ + ^ — o sec"o; cos 2

K7 sec" x \

*-&?-•«**'»*)<

186. It is required now to ascertain the limits between which
the integral -^3— , I [d(sQc"x) cos 2nix] which completes the series

is included. To form this integral an infinity of values must
be given to the arc x, from 0, the limit at which the integral
begins, up to oc, which is the final value of the arc ; for each one
of these values of x the value of the differential d (sec" x) must
be determined, and that of the factor cos 2mx, and all the partial
products must be added : now the variable factor cos 2mx is
necessarily a positive or negative fraction; consequently the
integral is composed of the sum of the variable values of the
differential fZ(scc".r), multiplied respectively by these fractions.

152 THEORY OF HEAT. [CHAP. III.

The total value of tlie integral is then less than the sum of the
differentials d (sec'7 a?), taken from x = 0 up to or, and it is greater
than this sum taken negatively : for in the first case we replace
tlie variable factor cos 2mx by the constant quantity 1, and in
the second case we replace this factor by — 1 : now the sum of
the differentials d (sec" x), or which is the same thing, the integral
{d (sec" x), taken from x = 0, is sec" x — sec'' 0 ; sec" x is a certain
function of x, and sec"0 is the value of this function, taken on
the supposition that the arc x is nothing.

The integral required is therefore included between
+ (sec"*e — sec" 0) and — (sec" x — sec" 0) ;

that is to say, representing by k an unknown fraction positive or
negative, we have always

/ {d (sec" x) cos 2mx] = k (sec" x — sec" 0).
Thus we obtain the equation

2u — c — — sec x cos 2mx + — — - sec' x sin Zmx + — — 3 sec" x cos Imx
'2m 2m 2°ra8

in which the quantity ^ — 3 (sec" x — sec" 0) expresses exactly the

.- fib

sum of all the last terms of the infinite series.

187. If we had investigated two terms only we should have
had the equation

I i jc

2t/ = c-~— sec x cos Zmx + -^—r, sec' x sin 2mx + -^—z (sec' x- sec'O).

*«///' _ y/6 ^ 7/&

From this it follows that we can develope the value of y in as
many terms as we wish, and express exactly the remainder of
the series ; we thus find the set of equations

1 ^k

*2i/ = c — x— sec x cos 2mx-^—t (sec x — sec 0),
9 2 in %'m

2 y — c — x— sec x cos 2mx+ ^— = sec' x sin 2mx \ ^7—., (sec' x — sec' 0),
2??^ 2 m 2 m v

2 y = c -- sec x cos %mx+ TT — 5 sec' # sin 2m^ 4- ^ — 5 sec" x cos 2m#
^w 2w 2 m

f " '' /\\

Hr n»~s (sec a;— sec 0).

1. /72-

SECT. III.] LIMITS OF THE VARIABLE. 153

The number k which enters into these equations is not the
same for all, and it represents in each one a certain quantity
which is always included between 1 and — 1 ; m is equal to the
number of terms of the series

cos x — - cos 3# + - cos 5x — . . . — — — -^ cos (2m — 1) xt
o 5 ~ili — 1

whose sum is denoted by y.

188. These equations could be employed if the number m
were given, and however great that number might be, we could
determine as exactly as we pleased the variable part of the value
of y. If the number m be infinite, as is supposed, we consider
the first equation only; and it is evident that the two terms
which follow the constant become smaller and smaller; so that
the exact value of 2y is in this case the constant c; this constant
is determined by assuming x = 0 in the value of y, whence we
conclude

-- = COS X — •= COS Sx + - COS DX — ;= COS 7# + T: COS 9.E — &C.

4 3 o 7 9

It is easy to see now that the result necessarily holds if the
arc x is less than \ir. In fact, attributing to this arc a definite
value X as near to JTT as we please, we can always give to in

a value so great, that the term - — (sec a; — sec 0), which completes

the series, becomes less than any quantity whatever ; but the
exactness of this conclusion is based on the fact that the term
sec x acquires no value which exceeds all possible limits, whence
it follows that the same reasoning cannot apply to the case in
which the arc x is not less than JTT.

The same analysis could be applied to the series which express
the values of Ja?, log cos x, and by this means we can assign
the limits between which the variable must be included, in order
that the result of analysis may be free from all uncertainty ;
moreover, the same problems may be treated otherwise by a
method founded on other principles1.

189. The expression of the law of fixed temperatures in
a solid plate supposed the knowledge of the equation

1 Cf. De Morgan's Eiff. and Int. Calculus, pp. 605 — 609. [A. F.]

154 THEORY OF HEAT. [CHAP. III.

TT 1 1 1 - I

— = cos x — ;r cos 3x -f -z cos 5# — = cos / # + g cos 9u; — &c.

A simpler method of obtaining this equation is as follows :

If the sum of two arcs is equal to JTT, a quarter of the
circumference, the product of their tangent is 1; we have there
fore in general

i - TT — arc tan u -f arc tan -

a

the symbol arc tan u denotes the length of the arc whose tangent
is u, and the series which gives the value of that arc is well

known ; whence we have the following result :

If now we write e^"1 instead of u in equation (c), and in equa
tion (d), we shall have

I /

- TT = arc tan ex^~L + arc tan e~x^ ~lj

and j TT = cos x — •= cos ox + -„- cos ox — ^ cos 7x -}- -r- cos 9*i' — &c.
4 o o / 9

The series of equation (d) is always divergent, and that of
equation (b) (Art. 180) is always convergent; its value is JTT
or — ITT.

SECTION IV.

General solution.

190. We can now form the complete solution of the problem
which we have proposed ; .for the coefficients of equation (b)
(Art. 1G9) being determined, nothing remains but to substitute
them, and we have

^ .= e~x cos y - -- e~"x cos 3y 4- - e~Bx cos 5y - ^ e"7'r cos 7.y + &c....(a).

SECT. IV.] COEXISTENCE OF PARTIAL STATES. 1,55

This value of v satisfies the equation -j—t + -^ = 0 ; it becomes

nothing when we give to y a value equal to \TT or — JTT ; lastly,
it is equal to unity when x is nothing and y is included between
— ^TT and + |TT. Thus all the physical conditions of the problem
are exactly fulfilled, and it is certain that, if we give to each
point of the plate the temperature which equation (a) deter
mines, and if the base A be maintained at the same time at the
temperature 1, and the infinite edges B and C at the tempera
ture 0, it would be impossible for any change to occur in the
system of temperatures.

191. The second member of equation (a) having the form
of an exceedingly convergent series, it is always easy to deter
mine numerically the temperature of a point whose co-ordinates
os and y are known. The solution gives rise to various results
which it is necessary to remark, since they belong also to the
general theory.

If the point m, whose fixed temperature is considered, is very
distant from the origin A, the value of the second member of
the equation (a) will be very nearly equal to e~x cos y • it reduces
to this term if x is infinite.

4
The equation v = - e~x cos y represents also a state of the

solid which would be preserved without any change, if it were
once formed ; the same would be the case with the state repre-

4

sented by the equation v — ^— e 3x cos %y, and in general each

O7T

term of the series corresponds to a particular state which enjoys
the same property. All these partial systems exist at once in
that which equation (a) represents ; they are superposed, and
the movement of heat takes place with respect to each of them
as if it alone existed. In the state which corresponds to any
one of these terms, the fixed temperatures of the points of the
base A differ from one point to another, and this is the only con
dition of the problem which is not fulfilled ; but the general state
which results from the sum of all the terms satisfies this special
condition.

According as the point whose temperature is considered is

156 THEORY OF HEAT. [CHAP. III.

more distant from the origin, the movement of heat is less com
plex : for if the distance x is sufficiently great, each term of
the series is very small with respect to that which precedes it,
so that the state of the heated plate is sensibly represented by
the first three terms, or by the first two, or by the first only,
for those parts of the plate which are more and more distant
from the origin.

The curved surface whose vertical ordinate measures the
fixed temperature v, is formed by adding the ordinates of a
multitude of particular surfaces whose equations are

^ = e* cos y, 7~ = - K3* cos 3# ^ =^"5* cos 5 yt &c.

The first of these coincides with the general surface when x
is infinite, and they have a common asymptotic sheet,

If the difference v — vl of their ordinates is considered to be
the ordinate of a curved surface, this surface will coincide, when x
is infinite, with that whose equation is ^irv2 = — ±e~Zx cos 3y. All
the other terms of the series produce similar results.

The same results would again be found if the section at the
origin, instead of being bounded as in the actual hypothesis by
a straight line parallel to the axis of y, had any figure whatever
formed of two symmetrical parts. It is evident therefore that
the particular values

ae~x cos y, le~3x cos 3y, ce~5x cos 5y, &c.,

have their origin in the physical problem itself, and have a
necessary relation to the phenomena of heat. Each of them
expresses a simple mode according to which heat is established
and propagated in a rectangular plate, whose infinite sides retain
a constant temperature. The general system of temperatures
is compounded always of a multitude of simple systems, and the
expression for their sum has nothing arbitrary but the coeffi
cients a, b, c, d, &c.

192. Equation (a) may be employed to determine all the
circumstances of the permanent movement of heat in a rect
angular plate heated at its origin. If it be asked, for example,
what is the expenditure of the source of heat, that is to say,

SECT. IV.] EXPENDITURE OF THE SOURCE OF HEAT. 157

what is the quantity which, during a given time, passes across
the base A and replaces that which flows into the cold masses
B and (7; we must consider that the flow perpendicular to the

axis of y is expressed by —K^-. The quantity which during
the instant dt flows across a part dy of the axis is therefore

and, as the temperatures are permanent, the amount of the flow,
during unit of time, is —K-j-dy. This expression must be

integrated between the limits y = — \-rr and y = 4- JTT, in order
to ascertain the whole quantity which passes the base, or which
is the same thing, must be integrated from y — 0 to y = JTT, and

the result doubled. The quantity -,- is a function of x and y,

CLJO

in which x must be made equal to 0, in order that the calculation
may refer to the base A, which coincides with the axis of y. The
expression for the expenditure of the source of heat is there

fore 2lf—Kj-dy}. The integral must be taken from y = Q to

y = ITT ; if, in the function -j- , x is not supposed equal to 0,

but x = x, the integral will be a function of x which will denote
the quantity of heat which flows in unit of time across a trans
verse edge at a distance x from the origin.

193. If we wish to ascertain the quantity of heat which,
during unit of time, passes across a line drawn on the plate

parallel to the edges B and C, we employ the expression — K -j~ ,

j
and, multiplying it by the element dx of the line drawn, integrate

with respect to x between the given boundaries of the line ; thus

the integral If— K -j- dx) shews how much heat flows across the
A dy J

whole length of the line ; and if before or after the integration
we make y = \TT, we determine the quantity of heat which, during
unit of time, escapes from the plate across the infinite edge C.
We may next compare the latter quantity with the expenditure

158 THEORY OF HEAT. [CHAP. III.

of the source of heat; for the source must necessarily supply
continually the heat which flows into the masses B and C. If
this compensation did not exist at each instant, the system of
temperatures would be variable.

194. Equation (a) gives
— K -7V = — (e~x cos y — e~sx cos 3y + e~r°x cos oy - e~"x cos 7y + &c.);

CLJC 7T

multiplying by dy, and integrating from 2/ = 0, we have

- ( e~x sin y — - e~5x sin 3y + - e~5x sin oy — ^ e~7* sin 7y -f &c. ] .

If y be made = JTT, and the integral doubled, we obtain

87T/ 1 _sv 1 _.x 1
— \e 4-^e fg« +7

as the expression for the quantity of heat which, during unit of
time, crosses a line parallel to the base, and at a distance x from
that base.

From equation (a) we derive also
— K -j- = -- (e~x sin y — e~Bx sin Sy + e~zx sin oy — e~lx sin 7y + &c.) :

hence the integral I— K I -j- j dx, taken from x = 0, is
r {(1 - e~") sin ?/ - (1 - e"3:") sin 3?/ + (1 - e"'*) sin 5y

If this quantity be subtracted from the value which it assumes
when x is made infinite, we find

- ( e~x sin y — - e~3x sin Sy + ^ e~*x sin oy — &c. ) ;

7T \ O O /

and, on making ?/ = j7r, we have an expression for the whole
quantity of heat which crosses the infinite edge C, from the
point whose distance from the origin is x up to the end of the
plate ; namely,

SECT. IV.] PERMANENT STATE OF THE RECTANGLE. 159

which is evidently equal to half the quantity which in the same
time passes beyond the transverse line drawn on the plate at
a distance x from the origin. We have already remarked that
this result is a necessary consequence of the conditions of the
problem ; if it did not hold, the part of the plate which is
situated beyond the transverse line and is prolonged to infinity
would not receive through its base a quantity of heat equal to
that which it loses through its two edges ; it could not therefore
preserve its state, which is contrary to hypothesis.

195. As to the expenditure of the source of heat, it is found
by supposing x = 0 in the preceding expression ; hence it assumes
an infinite value, the reason for which is evident if it be remarked
that, according to hypothesis, every point of the line A has and
retains the temperature 1 : parallel lines which are very near
to this base have also a temperature very little different from
unity: hence, the extremities of all these lines contiguous to
the cold masses E and C communicate to them a quantity of
heat incomparably greater than if the decrease of temperature
were continuous and imperceptible. In the first part of the
plate, at the ends near to B or (7, a cataract of heat, or an
infinite flow, exists. This result ceases to hold when the distance
x becomes appreciable.

196. The length of the base has been denoted by TT. If we
assign to it any value 2^, we must write \ifj instead of y, and

77" 'X1

multiplying also the values of a? by ~ , we must write JTT -.-
instead of x. Denoting by A the constant temperature of the
base, we must replace v by -r . These substitutions being made
in the equation (a), we have
v = — ( e""** cos --.'. — - e ~u cos 3 4~, + - e~ ~M cos 5 4,7

7T \ 'J.L 0 Z.I 3 £1

-^6 « cos7^ + &c.J (£).

This equation represents exactly the system of permanent
temperature in an infinite rectangular prism, included between
two masses of ice B and (7, and a constant source of heat.

160 THEORY OF HEAT. [CHAP. III.

197. It is easy to see either by means of this equation, or
from Art. 171, that heat is propagated in this solid, by sepa
rating more and more from the origin, at the same time that it
is directed towards the infinite faces B and G. Each section
parallel to that of the base is traversed by a wave of heat which
is renewed at each instant with the same intensity: the intensity
diminishes as the section becomes more distant from the origin.
Similar movements are effected with respect to any plane parallel
to the infinite faces; each of these planes .is traversed by a con
stant wave which conveys its heat to the lateral masses.

The developments contained in the preceding articles would
be unnecessary, if we had not to explain an entirely new theory,
whose principles it is requisite to fix. With that view we add
the following remarks.

198. Each of the terms of equation (a) corresponds to only
one particular system of temperatures, which might exist in a
rectangular plate heated at its end, and whose infinite edges are
maintained at a constant temperature. Thus the equation
v = e~x cos y represents the permanent temperatures, when the
points of the base A are subject to a fixed temperature, denoted
by cos y. We may now imagine the heated plate to be part of a
plane which is prolonged to infinity in all directions, and denoting
the co-ordinates of any point of this plane by x and y, and the
temperature of the same point by vt we may apply to the entire
plane the equation v = e~x cos y ; by this means, the edges B and
G receive the constant temperature 0 ; but it is not the same
with contiguous parts BB and CO ; they receive and keep lower
temperatures. The base A has at every point the permanent
temperature denoted by cos y, and the contiguous parts A A have
higher temperatures. If we construct the curved surface whose
vertical ordinate is equal to the permanent temperature at each
point of the plane, and if it be cut by a vertical plane passing
through the line A or parallel to that line, the form of the section
will be that of a trigonometrical line whose ordinate represents
the infinite and periodic series of cosines. If the same curved
surface be cut by a vertical plane parallel to the axis of x, the
form of the section will through its whole length be that of a
logarithmic curve.

SECT. IV.] FINAL PERMANENT STATE. 1G1

199. By this it may be seen how the analysis satisfies the
two conditions of the hypothesis, which subjected the base to a
temperature equal to cosy, and the two sides B and C to the
temperature 0. When we express these t\vo conditions we solve
in fact the following problem : If the heated plate formed part of
an infinite plane, what must be the temperatures at all the points
of the plane, in order that the system may be self-permanent, and
that the fixed temperatures of the infinite rectangle may be those
which are given by the hypothesis ?

We have supposed in the foregoing part that some external
causes maintained the faces of the rectangular solid, one at the
temperature 1, and the two others at the temperature 0. This
effect may be represented in different manners; but the hypo
thesis proper to the investigation consists in regarding the prism
as part of a solid all of whose dimensions are infinite, and in deter
mining the temperatures of the mass which surrounds it, so that
the conditions relative to the surface may be always observed.

200. To ascertain the system of permanent temperatures in
a rectangular plate whose extremity A is maintained at the tem
perature 1, and the two infinite edges at the temperature 0, we
might consider the changes which the temperatures undergo,
from the initial state which is given, to the fixed state which is
the object of the problem. Thus the variable state of the solid
would be determined for all values of the time, and it might then
be supposed that the value was infinite.

The method which we have followed is different, and conducts
more directly to the expression of the final state, since it is
founded on a distinctive property of that state. We now proceed
to shew that the problem admits of no other solution than that
which we have stated. The proof follows from the following
propositions.

201. If we give to all the points of an infinite rectangular
plate temperatures expressed by equation (2), and if at the two
edges B and C we maintain the fixed temperature 0, whilst the
end A is exposed to a source of heat which keeps all points of the
line A at the fixed temperature 1; no change can happen in the

state of the solid. In fact, the equation -y-a + -=-$ = 0 being

F. H. n

162 THEORY OF HEAT. [CHAP. III.

satisfied, it is evident (Art. 170) that the quantity of heat which
determines the temperature of each molecule can be neither
increased nor diminished.

The different points of the same solid having received the
temperatures expressed by equation (a) or v = <f*(x,y), suppose
that instead of maintaining the edge A at the temperature 1, the
fixed temperature 0 be given to it as to the two lines B and C ';
the heat contained in the plate BAG will flow across the three
edges A, B, C, and by hypothesis it will not be replaced, so that
the temperatures will diminish continually, and their final and
common value will be zero. This result is evident since the
points infinitely distant from the origin A have a temperature
infinitely small from the manner in which equation (a) was
formed.

The same effect would take place in the opposite direction, if
the system of temperatures were v = — (f> (x, y), instead of being
v = (j) (x, y) ; that is to say, all the initial negative temperatures
would vary continually, and would tend more and more towards
their final value 0, whilst the three edges A, B, C preserved the
temperature 0.

202. Let v = $ (x, y) be a given equation which expresses
the initial temperature of points in the plate BA C, whose base A
is maintained at the temperature 1, whilst the edges B and C
preserve the temperature 0.

Let v = F(x, y} be another given equation which expresses
the initial temperature of each point of a solid plate BAG exactly
the same as the preceding, but whose three edges B, A, G are
maintained at the temperature 0.

Suppose that in the first solid the variable state which suc
ceeds to the final state is determined by the equation v = (f>(x, y, t\
t denoting the time elapsed, and that the equation v = <3> (x, y, t)
determines the variable state of the second solid, for which the
initial temperatures are F(x, y}.

Lastly, suppose a third solid like each of the two preceding:
let v =f(x, y) + F(xt y) be the equation which represents its
initial state, and let 1 be the constant temperature of the base
Ay 0 and 0 those of the two edges B and C.

SECT. IV.] SUPERPOSITION OF EFFECTS. 163

We proceed to shew that the variable state of the third solid
is determined by the equation v = (f>(x, y, t} + <!>(#, y, £)„

In fact, the temperature of a point m of the third solid varies,
because that molecule, whose volume is denoted by M, acquires
or loses a certain quantity of heat A. The increase of tempera
ture during the instant dt is

the coefficient c denoting the specific capacity with respect to
volume. The variation of the temperature of the same point in

the first solid is ~^ dt, and —^dt in the second, the letters

d and D representing the quantity of heat positive or negative
which the molecule acquires by virtue of the action of all the
neighbouring molecules. Now it is easy to perceive that A
is equal to d + D. For proof it is sufficient to consider the
quantity of heat which the point m receives from another point
m' belonging "to the interior of the plate, or to the edges which
bound it.

The point ??&,, whose initial temperature is denoted by fv
transmits, during the instant dt, to the molecule m, a quantity of
heat expressed by qj.f^ —f)dtt the factor ql representing a certain
function of the distance between the two molecules. Thus the
whole quantity of heat acquired by in is 'S.q^f^—f^jdt, the sign
2 expressing the sum of all the terms which would be found
by considering the other points mz, m5, ???4 &c. which act on m ;
that is to say, writing q2,/2 or ^3,/3, or q^ /4 and so on, instead of
qv fv In the same manner '^ql(Fl — F)dt will be found to be
the expression of the whole quantity of heat acquired by the
same point in of the second solid ; and the factor ql is the same
as in the term 2$\C/i —f)dt, since the two solids are formed of
the same matter, and the position of the points is the same; we
have then

d = *?,(./; -/)* and D = Sfc(F, - F)dt,
For the same reason it will be found that

11—2

, A d T)

hence A = d + D and -^ = —;, -f -j-, .

cM cM cM

164 THEORY OF HEAT. [CHAP. III.

It follows from this that the molecule m of the third solid
acquires, during the instant dt, an increase of temperature equal
to the sum of the two increments which the same point would
have gained in the two first solids. Hence at the end of the
first instant, the original hypothesis will again hold, since any
molecule whatever of the third solid has a temperature equal
to the sum of those which it has in the two others. Thus the
same relation exists at the beginning of each instant, that is to
say, the variable state of the third solid can always be represented
by the equation

203. The preceding proposition is applicable to all problems
relative to the uniform or varied movement oinea^7 It shews
that the movement can always be decomposed into several others,
each of which is effected separately as if it alone existed. This
superposition of simple effects is one of the fundamental elements
in the theory of heat. It is expressed in the investigation, by
the very nature of the general equations, and derives its origin
from the principle of the communication of heat.

Let now v — <£ (x, y] be the equation (a) which expresses the
permanent state of the solid plate BAG, heated at its end A, and
whose edges B and C preserve the temperature i; the initial state
of the plate is such, according to hypothesis, that all its points
have a nul temperature, except those of the base A, whose tem
perature is 1. The initial state can then be considered as formed
of two others, namely : a first, in which the initial temperatures are
— (j>(x, y), the three edges being maintained at the temperature 0,
and a second state, in which the initial temperatures are + <j>(x,y),
the two edges B and C preserving the temperature 0, and the
base A the temperature 1; the superposition of these two states
produces the initial state which results from the hypothesis. It
remains then only to examine the movement of heat in each one
of the two partial states. Now, in the second, the system of tem
peratures can undergo no change ; and in the first, it has been
remarked in Article 201 that the temperatures vary continually,
and end with being nul. Hence the final state, properly so called,
is that which is represented by v = $ (x, y] or equation (a).

SECT. IV.] THE FINAL STATE IS UNIQUE. 165

If this state were formed at first it would be self-existent, and
it is this property which has served to determine it for us. If the
solid plate be supposed to be in another initial state, the differ
ence between the latter state and the fixed state forms a partial
state, which imperceptibly disappears. After a considerable time,
the difference has nearly vanished, and the system of fixed tem
peratures has undergone no change. Thus the variable temper
atures converge more and more to a final state, independent of
the primitive heating.

204. We perceive by this that the final state is unique; for,
if a second state were conceived, the difference between the
second and the first would form a partial state, which ought to be
self-existent, although the edges A, B, C were maintained at the
temperature 0. Now the last effect cannot occur; similarly if we
supposed another source of heat independent of that which flows
from the origin A] besides, this hypothesis is not that of the
problem we. have treated, in which the initial temperatures are
nul. It is evident that parts very distant from the origin can
only acquire an exceedingly small temperature.

Since the final state which must be determined is unique, it
follows that the problem proposed admits no other solution than
that which results from equation (a). Another form may be
given to this result, but the solution can be neither extended nor
restricted without rendering it inexact.

The method which we have explained in this chapter consists
in formnig fiFst very simple particular values, which agree with
the .problem, and in rendering the solution more general, to the
intent that v or </> (as, y) may satisfy three conditions, namely :

It is clear that the contrary order might be followed, and the
solution obtained would necessarily be the same as the foregoing.
We shall not stop over the details, which are easily supplied,
when once the solution is known. We shall only give in the fol»
lowing section a remarkable expression for the function </> (x, y]
whose value was developecTm a convergent series in equation (a).

166 THEORY OF HEAT. [CHAP. III.

SECTION V.

Finite expression of the result of the solution^

205. The preceding solution might be deduced from the

d2v d*v
integral of the equation -y~2 + -3-3 = O,1 which contains imaginary

quantities, under the sign of the arbitrary functions. We shall
confine ourselves here to the remark that the integral

v=<!>(x+yj -T) +^r(x- W^T),
has a manifest relation to the value of v given by the equation

-T- = e~x cos y — ^ e~Zx cos 3y -f ^ e~5x cos oy — &c.
4 o 5

In fact, replacing the cosines by their imaginary expressions,
we have

- &c.
3 o

The first series is a function of x — yJ—\, and the second
series is the same function of x + yj — 1.

Comparing these series with the known development of arc tan z
in functions of z its tangent, it is immediately seen that the first
is arc tan eif**'f3r\ and the second is arc tan e'^^ ; thus
equation (a) takes the finite form

~ = arc tan e-(x+v^ + arc tan e-<*-»v=r>'

In this mode it conforms to the general integral

v = <t>(x + yj~\) + ^(x-yj~^l) ......... (A),

the function $ (z) is arc tan e~", and similarly the function i|r (z).
1 D. F. Gregory derived the solution from the form

Cumb. Math. Journal, Vol. I. p. 105. [A. F.]

SECT. V.] FINITE EXPRESSION OF THE SOLUTION. 167

If in equation (B) we denote the first term of the second mem
ber by p and the second by q, we have

, N tan p -f tan a 2e~x cos y 2 cos y
whence tan (p + g) or - f - — - == - - txf = — - ^ ;

1 — tan p tan q 1 — e e — e

1 /2 cos y\ .f

whence we deduce the equation -TTV = arc tan ( -- - _-} ...(..(G).

A \& —e J

This is the simplest form under which the solution of the
problem can be presented.

206. This value of v or c/> (x, y) satisfies the conditions relative
to the ends of the solid, namely, (/> (x, ± JTT) = 0, and (j> (0, y} = 1 ;

70 72

it satisfies also the general equation — + —-2 = 0, since equa

tion ((7) is a transformation of equation (B). Hence it represents
exactly the system of permanent temperatures ; and since that
state is unique, it is impossible that there should be any other
solution, either more general or more restricted.

The equation (C) furnishes, by means of tables, the value of
one of the three unknowns v, x, y} when two of them are given; it
very clearly indicates the nature of the surface whose vertical
ordinate is the permanent temperature of a given point of the
solid plate. Finally, we deduce from the same equation the values

of the differential coefficients -=- and -y- which measure the velo-

ax ay

city with which heat flows in the two orthogonal directions ; and
we consequently know the value of the flow in any other direction.

These coefficients are expressed thus,

dx
dv

It may be remarked that in Article 194 the value of -j- , and
that of -j- are given by infinite series, whose sums may be easily

168 THEORY OF HEAT. [CHAP. III.

found, by replacing the trigonometrical quantities by imaginary

exponentials. We thus obtain the values of -3- and -r- which

ace ay

we have just stated.

The problem which we have now dealt with is the first which
we have solved in the theory of heat, or rather in that part of
the theory which requires the employment of analysis. It
furnishes very easy numerical applications, whether we make
use of the trigonometrical tables or convergent series, and it
represents exactly all the circumstances of the movement of
heat. We pass on now to more general considerations.

SECTION VI.

Development of an arbitrary function in trigonometric series.

207. The problem of the propagation of heat in a rect-

d2v d2v
angular solid has led to the equation -y-g + -=-£ = 0 ; and if it

be supposed that all the points of one of the faces of the solid
have a common temperature, the coefficients a, b, c, d} etc. cf
the series

a cos x + b cos 3x + c cos 5# 4- d cos 7x + ... &c.,

must be determined so that the value of this function may be
equal to a constant whenever the arc x is included between — JTT
and + JTT. The value of these coefficients has just been assigned;
but herein we have dealt with a single case only of a more general
; problem, which consists in developing any function whatever in
an infinite series of sines or cosines of multiple arcs. This
problem is connected with the theory of partial differential
equations, and has been attacked since the origin of that analysis.
It was necessary to solve it, in order to integrate suitably the
equations of the propagation of heat; we proceed to explain
the solution.

We shall examine, in the first place, the case in which it is
required, to reduce into a series of sines of multiple arcs, a
function whose development contains only odd powers of the

SECT. VI.] SERIES OF SINES OF MULTIPLE ARCS. 160

variable. Denoting such a function by <£ (x), we arrange the
equation

(j) (x) = a sin x + b sin 2x -f c sin 3x + d sin 4<x -f . . . &c.,
in which it is required to determine the value of the coefficients
a, b, c, d, &c. First we write the equation ^

<^(^) = ^Xo) + |V'Xo)+^f'Xo) + ^^o) + ^xo)+..^W M

If. !_ l_ 2.

in which <£'(0), <£"(0), ^'"(0), <£lv(0), &c. denote the values taken
by the coefficients

(x)

* ' c

dx dx* ' da? ' dx

when we suppose x — 0 in them. Thus, representing the develop
ment according to powers of x by the equation

we have <j> (0) = 0, and <f>' (0) = A,

&c. &c.

If now we compare the preceding equation with the equation
<j)(x) = a sin x + b sin 2x + c sin 3# + J sin 4<x + e sin 5^ -|- &c.,

developing the second member with respect to powers of x, we
have the equations

A = a + 2Z> + 3c + 4d + 5e + &c.,
£ = a + 236 + 33c + tfd + 53e + &c.,
(7= a + 25^ + 35c + 45cZ + 55e + &c.,
D = a + 276 + 37c + 47d + 57e + &c.,

These equations serve to find the coefficients a, b, c, d, e,
&c., whose number is infinite. To determine them, we first re
gard the number of unknowns as finite and equal to m ; thus
we suppress all the equations which follow the first m equations,

170 THEORY OF HEAT. [CHAP. III.

and we omit from each equation all the terms of the second
member which follow the first m terms which we retain. The
whole number m being given, the coefficients a, b, c, d, e, &c. have
fixed values which may be found by elimination. Different
values would be obtained for the same quantities, if the number
of the equations and that of the unknowns were greater by one
unit. Thus the value of the coefficients varies as we increase
the number of the coefficients and of the equations which ought
to determine them. It is required to find what the limits are
towards which the values of the unknowns converge continually
as the number of equations increases. These limits are the true
values of the unknowns which satisfy the preceding equations
when their number is infinite.

208. We consider then in succession the cases in which we
should have to determine one unknown by one equation, two
unknowns by two equations, three unknowns by three equations,
and so on to infinity.

Suppose that we denote as follows different systems of equa
tions analogous to those from which the values of the coefficients
must be derived :

a^ = A^ aa + 262 = Aa, a3 + 2&3 + 3c3 = Az,

3c4

3c5

&c. &c ......... . ................ (b).

SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 171

If now we eliminate the last unknown e& by means of the
five equations which contain A&, B&) C5, D5, E.., &c., we find

a. (52 - I2) + 2\ (52 - 22) + 3\ (52 - 32)
a5 (52 - I2) + 25£5 (52 - 22) + 35c5 (52 - 32)
o5 (52 - I2) + 27£5 (52 - 22) + 3V5 (52 - 32)

We could have deduced these four equations from the four
which form the preceding system, by substituting in the latter
instead of

c4, (52-32)c5,

rf4f (52-42)c/5;

and instead of At, DzA^ — Bb,

Bt, 5'Jf-C-.,

C4I 5' (7. -/>.,

By similar substitutions we could always pass from the case
which corresponds to a number m of unknowns to that which
corresponds to the number m-f-1. Writing in order all the
relations between the quantities which correspond to one of the
cases and those which correspond to the following case, we shall
have

= cs (52 - 32), rf4 = rf5(52-42),

&c ............................ (c).

172 THEORY OF HEAT. [CHAP. III.

We have also

&c. &c .............................. (d).

From equations (c) we conclude that on representing the un
knowns, whose number is infinite, by a, b, c, d, e, &c., we must
have

a —

(3* - 22) (42 - 22) (52 - 22) (62 - 22) . . . '

~ (4a - 32) (52 - 32) (62 - 32) (T - 32) . . . '

d = (5* _ 4«) (G2 - 42) (T - 42) (82 - 42) . . . '

&c. &c (e).

209. It remains then to determine the values of alt 62, c8,
d4, ee, &c. ; the first is given by one equation, in which A± enters;
the second is given by two equations into which A2BZ enter; the
third is given by three equations, into which A3B3C3 enter ; and
so on. It follows from this that if we knew the values of

A19 A2B2, A3B3C3, Af^CJ),..., &c.,

we could easily find ax by solving one equation, a2&2 by solving
two equations, a3b3c3 by solving three equations, and so on : after
which we could determine a, b} c, d, e, &c. It is required then
to calculate the values of

..., &c,

by means of equations (d). 1st, we find the value of A2 in
terms of A% and 52; 2nd, by two substitutions we find this value
of A1 in terms of A3B3C3; 3rd, by three substitutions we find the

SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 173

same value of Al in terms of J4J54(74Z)4, and so on. The successive
values of A are

A, = A£\ 32. 42 - B, (22. 32 + 22. 42 + 32. 42) + <74 (22 + 32 + 42) - D4,

^^J^2^2^2^2-^^22-82-4^22-32-5^22-42-5^32-42-52)
+ C6 (22. 32 + 22. 42 + 22.52 + 32.42 + 32.52 + 42.52)
- Db (22 + 32 + 42 + 52) + E6, &c.,

the law of which is readily noticed. The last of these values,
which is that which we wish to determine, contains the quantities
A, B, C, D, E, &c., with an infinite index, and these quantities
are known ; they are the same as those which enter into equa
tions (a).

Dividing the ultimate value of A: by the infinite product

22.32.42.52.62...&c.,
we have

" D (.2*. 3". 4" + 2". 3". 5a + 3". 4". 5" + &C7
E '.S'.^.o1 + ¥^~4\ff + &C') + &C'

The numerical coefficients are the sums of the products which
could be formed by different combinations of the fractions

1 i i i i Ac

I2' 2" 3" 52' 6*"

after having removed the first fraction p. If we represent

the respective sums of products by Plf Qx, R^ Slt TI} ... &c., and
if we employ the first of equations (e) and the first of equa
tions (6), we have, to express the value of the first coefficient a,
the equation

22.32.42.52...

CQl - DRV + ESl - &c.,

174 THEORY OF HEAT. [CHAP. ITT.

now the quantities Plt Qlf Elt Slt T^... &c. may be easily deter
mined, as we shall see lower down ; hence the first coefficient a
becomes entirely known.

210. We must pass on now to the investigation of the follow
ing coefficients b, c, d, e, &c., which from equations (e) depend on
the quantities 62, c3, d4, es, &c. For this purpose we take up
equations (6), the first has already been employed to find the
value of ffj, the two following give the value of 62, the three
following the value of C3, the four following the value of d4, and
so on.

On completing the calculation, we find by simple inspection
of the equations the following results for the values of 62, cs, r74,
&c.

3c3 (I2 - 32) (22 - 32) = A3l2. 22 - Bz (I2 + 22) + <73,
4<Z4(l2-42)(22-42)(32-42)

= .44l2. 22. 32-^4 (I2. 22 + I2. 32 + 22.32) + C4 (12 + 22-f 32) -7>4,
&c.

It is easy to perceive the law which these equations follow ;
it remains only to determine the quantities AnBn, A2B3C3,
A$f!v &c.

Now the quantities A.2B2 can be expressed in terms of A3B3C3,
the latter in terms of A4B4C4D4. For this purpose it suffices to
effect the substitutions indicated by equations (d) ; the successive
changes reduce the second members of the preceding equations
so as to contain only the AB CD, &c. with an infinite suffix,
that is to say, the known quantities ABCD, &c. which enter into
equations (a) ; the coefficients become the different products
which can be made by combining the squares of the numbers
1*2*3*4*5* to infinity. It need only be remarked that the first
of these squares I2 will not enter into the coefficients of the
value of at ; that the second 22 will not enter into the coefficients
of the value of b.2 ; that the third square 32 will be omitted only
from those which serve to form the coefficients of the value of c3 ;
and so of the rest to infinity. We have then for the values of

SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 175

t>2c3d4e5, &c, and consequently for those of bcde, £c., results entirely
analogous to that which we have found above for the value of
the first coefficient a^.

211. If now we represent by P2, Q,, Pz, S2, &c., the quantities

1+1+1+1.

I2 3* 4* 5*

1*. 32 I2. 42 I2. 52 32.

&c.,
which are formed by combinations of the fractions 1 , 1 , 1 ,

£2 , ^5 ... &c. to infinity, omitting ^ the second of these fractions
we have, to determine the value of b2, the equation

, - &c.

Representing in general by PnQnRnSn ... the sums of the
products which can be made by combining all the fractions

p > 2* > g2 > f , -^2 "- to infinity, after omitting the fraction 1

only; we have in general to determine the quantities alt 62, c
d4, es..., &c., the following equations:

A-BPl+CQl-DBl

., ,

^- . O . -T . O ...

A - £P2 + CQ2 - DR + ES - &c. = 2i ,-'"2?

»4-^=«

la.22.3*.5«.6«.. ~
&c.

176 THEORY OF HEAT. [CHAP. III.

212. If we consider now equations (e) which give the values
of the coefficients a, 6, c, d, &c., we have the following results :

(22 - I2) (32 - I2) (42 - I2) (52 - I2) ...
22.32.42.52...

= A-BP1+ CQi - DE, + ESi - &c.,

(I2 _ 22) (32-22) (42-22) (52-22)...
12.3*.42.52...

= A-BP,+ CQ. - DR2 + ES2 - &c.,

32) (22 - 32) (4* - 32) (5* - 32) . . .

I2. 22.42.52...

(1« _ 4«) (22 - 42) (32 - 42) (52 - 42) . . .
I2.2a.32.o2...

= A - BP, + 0§4 - D^4 + ^^4 - &c.,

&c.

Remarking the factors which are wanting to the numerators
and denominators to complete the double series of natural
numbers, we see that the fraction is reduced, in the first equation

11 22 33

to =- . o ; in the second to — s • T > m ^ne third to - . ^ ; in the

4 4

fourth to — -r . ^ ; so that the products which multiply a, 2&, 3c,

4c£, &c., are alternately ^ and — « • It is only required then to
find the values of P&E&, P&R&, P3Q3^3^3, &c.

To obtain them we may remark that we can make these
values depend upon the values of the quantities PQRST, &c.,
which represent the different products which may be formed

with the fractions ^ , ^> -&> T2> ^2> 7&> &c-» without omit-

1 L O TT O O

ting any.

With respect to the latter products, their values are given
by the series for the developments of the sine. We represent
then the series

SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 177

+ 12 02 + -1 2 A9 + 92 02+02 J2 + 02 42 + &(
J. . O l.T) Zi . O Zi.rr O.T

!&c
I2. 22. 3* I2. 22. 4* I2. 32. 42 22. 32. 42

12.22.32.42 2*.32.42.52 F.22.32.52 >
by P, Q, 5, 5, &c.

aj3 x5 x7
The series sin# = # — s + j^ — ?= + &c.

3 |o 7

furnishes the values of the quantities P, Q, E, S, &c. In fact, the
value of the sine being expressed by the equation

we have

1-g+|-|+&ft

Whence we conclude at once that

213. Suppose now that Pw, QB, 5B, /Sfn, &c., represent the
sums of the different products which can be made with the

fractions — 2 , ^ , ^ , -^ , ^ , &c., from which the fraction -=

± Z o TC O 71,

has been removed, n being any integer whatever ; it is required
to determine Pn, Qn, En, Sn, &c., by means of P, Q, E, S, &c. If
we denote by

the products of the factors
1-

V

\ H.

178 THEORY OF HEAT. [CHAP. III.

among which the factor ( 1 - 4) only has been omitted ; it follows
that on multiplying by (l - J^J the quantity

we obtain 1 - qP + (f Q - fR + q*S - &c.

This comparison gives the following relations :

&c.;

&c.

Employing the known values of P, Q, JR, ft and making ?i
equal to 1, 2, 3, 4, 5, &c. successively, we shall have the values of
P&RA, &c. ; those of P2QA^» &c- 5 those of P&3RA &c'

214 From the foregoing theory it follows that the values
of a, b, c, d, e, &c., derived from the equations

a + 26 + 3c + 4d + 5e + &c. = -4,
a + 236 + 33c + 43^ + 5se + &c. = #,
a + 256 + 35c + tfd + tfe + &c. = 0,
a + 276 + 37c + 47rf + 57e + &c. = D,
a + 29^ + 39c + W + 5°e + &c. = ^,
&c.,

SECT. VI.] VALUES OF THE COEFFICIENTS.

are thus expressed,

179

a- A B

' -

[7

1T* 1 7T6 1 7T4 1 7T2

~ + -

(7

-lzL6^!^4 l772^ x>\ ^

[9 22|7+24[5""26|3+2V~

D^-lzL4^!^..!^

Vg 32|5+34[3 3V
,F/7r_8_^7r_6 ITT* j. 7r2 .

1-1 2 * 6 8 "

-D^.l^4, l^.n
l7 42 5 + 44 3 4V

|3

&c.

215. Knowing the values of a, b, c, d, e, &c., we can substitute
them in the proposed equation

<£ (x) = a sin x + b sin 2# + c sin 3# + d sin 4;c + e sin ox + &c.,

and writing also instead of the quantities A, B, C, D, E, &c., their

12—2

180 THEORY OF HEAT. [CHAP. HI.

values £'(0), <J>"'(0), (£v(0), <£vii(0), <£lx(0), &c., we have the general
equation

jjf

+ &C.

We may make use of the preceding series to reduce into
a series of sines of multiple arcs any proposed function whose
development contains only odd powers of the variable.

216. The first case which presents itself is that in which
4> (as) = «?; we find then </>'(0) = 1, <£"'(0) = 0, <£v(°) = 0, &c., and so
for the rest, We have therefore the series

x on = sin x — n sin 2x + ^ sin 3x — -r sin 4# + &c.,
4j . " 2 o 4

which has been given by Euler.

If we suppose the proposed function to be x*, we shall have

<£' (0) = 0, f "(0) = [3, $ (0) = 0, </>«'((>) = 0, &o.,
which gives the equation

- a? = \7rz - -j= J sin x - (TT* - L= J sin 2cc -}- ^7r2 - -^J g sin 3ic -f &c.

(A).

SECT. VI.] DEVELOPMENTS IN SERIES OF SINES. 181

We should arrive at the same result, starting from the pre
ceding equation,

-x = sin x — ^ sin 2# + ^ sin 3x - -r sin 4# + &c.

A A 6 *f

In fact, multiplying each member by dx, and integrating, we
have

C — -r — cos x— ~a cos 2x -f ^ cos &£ — -rs cos 4# -f &c. ;

4 .Z o 4*

the value of the constant (7 is

a series whose sum is known to be ~ -^ . Multiplying by dx the

two members of the equation

ITT2 X*

2 -« — -T = co
and integrating we have

ITT2 X* 1 1

2 -« — -T = cos a; - ^2 cos 2x + -^ cos 3# - &c.,

If now we write instead of x its value derived from the
equation

^ # = sin a? — TT sin 2# + ^ sin 3# — -7 sin 4# + &c.,

we shall obtain the same equation as above, namely,

7T2

We could arrive in the same manner at the development in
series of multiple arcs of the powers x5, a?, x9, &c., and in general
every function whose development contains only odd powers of

the variable.

5-

217. Equation (A), (Art. 218), can be put under a simpler
form, which we may now indicate. We remark first, that part of
the coefficient of sin x is the series

*'(0) + V "(0) + #(0) + r (0) + &c,

182 THEORY OF HEAT. [CHAP. III.

which represents the quantity -(/>(TT). In fact, we have, in
general,

'(0)*|*"(0)+|*'

&c.

Now, the function <f>(x) containing by hypothesis only odd
powers, we must have <£(0) = 0, £"(0) = 0, </>iv(0) = 0, and so on.
Hence

<f) (x) = x(j)(Q) + TK <fi" (0) + p 0VW + <^c< j

a second part of the coefficient of sin x is found by multiplying
by — Q the series

<T'(0) + n>3 ^(0) + IF </>vli(0) + ^ ^lx(°) + &c->

whose value is - $' (TT}. We can determine in this manner the

7Tr

different parts of the coefficient of sin#, and the components of
the coefficients of sin 2#, sin 3x, sin 4<x, &c. We may employ for
this purpose the equations :

f (0) + *' "(0) + <^V(0) + &c. =

r (0) + ^(0) + &c. =

^'(°> ^ &c- = -

O 7T

SECT. VI.] DEVELOPMENTS IN SERIES OP SINES. 183

By means of these reductions equation (A) takes the following
form :

sn x

« - J f ' (TT) + J <£iv(7r) - J ^(TT) + &cj
- i sin 2* {</> (TT) - I <£" (TT) + 1 4>lv (TT) - 1 </>» + &c. J

sin 3* (/> (TT) - f (TT) + ^ (TT) - <^(TT) + &*

- sn * c W - ^'(T) + r W - ^ W + &

(B);

or this,
5

a?) = ^ (TT) •! sin x — — sin 2,r + — sin 3x — &c. h

— <t>" (TT) | sin ^ — ^ sin 2:c + ^ sin 3x — &c. [•

[ ^ o )

+ (/>IV (TT) -jsin x — -^ sin 2x + ^ sin 3o? — &c. ^

— c/)vl (TT) •! sin x — -^ sin 2x + ^? sin 3uC — &c. [•

+ &c. (C).

218. We can apply one or other of these formulas as often as
we have to develope a proposed function in a series of sines of
multiple arcs. If, for example, the proposed function is ex — e~*t
whose development contains only odd powers of x, we shall have

1 (F —. Q~* / 1 1 \

x TT — - — = f sin x — -^ sin 2# + — sin 3^ — &c. J

^ *Vu — (sin a; — ^ sin 2ic + ^ sin 3a; — &c. )

*% ! i

*t*3 + (sm ^ — «B sm 2ic + o5 sin 3x — &c. J
i

— ( sin x — yj sin 2x + ^ sin 3,£ — &c. J

184 THEORY OF HEAT. [CHAP. III.

Collecting the coefficients of sin x, sin 2x, sin 3#, sin 4*x, &c.,
I.i
have

and writing, instead of * + -* 7+ etc.. its value -, — - , we

n n* n5 tf ri* + 1

1 (e* — e x) _ sin x sin 2x sin 3#

271" e^-e^ ~1~11~ °'

We might multiply these applications and derive from them
several remarkable series. We have chosen the preceding example
because it appears in several problems relative to the propagation
of heat.

219. Up to this point we have supposed that the function
whose development is required in a series of sines of multiple
arcs can be developed in a series arranged according to powers
of the variable xt and that only odd powers enter into that
series. We can extend the same results to any functions, even
to those which are discontinuous and entirely arbitrary. To esta
blish clearly the truth of this proposition, we must follow the
analysis which furnishes the foregoing equation (B), and examine
what is the nature of the coefficients which multiply sin a?,

sin 2x, sin 3#, &c. Denoting by - the quantity which multiplies

•ftr

-sin nx in this equation when n is odd, and — s'mnx when n is
n n

even, we have

• a = <KT) - J *"« + J <f W - i *» + &C.

Hi Hi It/

Considering s as a function of TT, differentiating twice, and

1 d?s
comparing the results, we find s + -$ ~r-2 = </> (TT) ; an equation

ft Cvr/r

which the foregoing value of 5 must satisfy.

1 dzs

Now the integral of the equation s +-5 T~I = </> (#)> m which s

fft CLtjG

is considered to be a function of a?, is
s — a cos nx + b sin nx

4- n sin nx \ cos nx $ (x) dx — n cos nx I sin nx 0 (x) dx.

SECT. VI.] GENERAL FORMULA. 185

If n is an integer, and the value of x is equal to TT, we have
s = ± n \(f> (x) sinnxdx. The sign + must be chosen when n is

odd, and the sign — when that number is even. We must make
x equal to the semi-circumference TT, after the integration in
dicated; the result may be verified by developing the term

| (/> (x) sin nx dx, by means of integration by parts, remarking

that the function <£ (x) contains only odd powers of the vari
able x, and taking the integral from x = 0 to x = TT.
We conclude at once that the term is equal to

o

If we substitute this value of - in equation (B), taking the
sign + when the term of this equation is of odd order, and the
sign — when n is even, we shall have in general I $(x) sin nxdx

for the coefficient of sin?z#; in this manner we arrive at a very \
remarkable result expressed by the following equation :

7T(j>(x) = since I sin x$(x) dx + sin 2x /sin 2#<£ (x) dx+&c.
J J

in/ic lsini#<£ (x) dx + &c .............. f. (D), /

.

"sX

the second member will always give the development required
for the function </>(#), if we integrate from x = 0 to # = 7r.1

1 Lagrange had already shewn (Miscellanea Taurinensia, Tom. in., 1760,
pp. 260 — 1) that the function y given by the equation

y = 2 (iTV, sin Xr-rr AX) sin xir + 2 (5TVr sin 2XrTr AX) sin 2xir

r=l r=l

+ 2 (iT Yr sin 3Xr7r AX) sin 3xir + . . . + 2 (S^Yr sin nXrv AX ) sin nxir
receives the values F1} Y^, Y3...Yn corresponding to the values Xlt X2, X3...Xn of
x, where Xr = , and AX .

Lagrange however abstained from the transition from this summation-formula
to the integration-formula given by Fourier.

Cf. Riemann's Gcsammclte Mathcmatische Werke, Leipzig, 1876, pp. 218—220
of his historical criticism, Ucber die Darstellbarkeit einer Function durch eine
Trigonomctritche Reihe. [A. F.]

186 THEORY OF HEAT. [CHAP. III.

220. We see by this that the coefficients a, b, c, d, e,f, &c.,
which enter into the equation

5 Tr<p (x) — a sin x + b sin 2x + c sin 3x + d sin 4# + &c.,

and which we found formerly by way of successive eliminations,
are the values of definite integrals expressed by the general term

sin ix (j> (x) dx} i being the number of the term whose coefficient

is required. This remark is important, because it shews how even
entirely arbitrary functions may be developed in series of sines
of multiple arcs. In fact, if the function <£ (x) be represented
by the variable ordinate of any curve whatever whose abscissa
extends from x = 0 to x — TT, and if on the same part of the axis
the known trigonometric curve, whose ordinate is y — sin x, be
constructed, it is easy to represent the value of any integral
term. We must suppose that for each abscissa x, to which cor
responds one value of $ (a?), and one value of sin x, we multiply
the latter value by the first, and at the same point of the axis
raise an ordinate equal to the product $ (x) sin x. By this con
tinuous operation a third curve is formed, whose ordinates are
~those of the trigonometric curve, reduced in proportion to the
^ordinates of the arbitary curve which represents <£(#). This
done, the area of the reduced curve taken from x = 0 to X = TT
gives the exact value of the coefficient of sin#; and whatever
the given curve may be which corresponds to $ (#), whether we
can assign to it an analytical equation, or whether it depends on
110 regular law, it is evident that it always serves to reduce
in any manner whatever the trigonometric curve; so that the
area of the reduced curve has, in all possible cases, a definite
value, which is the value of the coefficient of sin x in the develop
ment of the function. The same is the case with the following

coefficient b, or /<£ (x) sin 2xdx.

In general, to construct the values of the coefficients a, b, c, d, &c.,
\\e must imagine that the curves, whose equations are

y = sin x, y = sin Zx, y = sin Sx, y = sin 4#, &c.,
have been traced for the same interval on the axis of x, from

SECT. VI.] VERIFICATION OF THE FORMULA. 187

x = 0 to x = TT ; and then that we have changed these curves by
multiplying all their ordinates by the corresponding ordinates of
a curve whose equation is y = <f>(x). The equations of the re
duced curves are

y = sin x cf> (x), y = sin 2x </> (x), y = sin 3x </> (x), &c.

The areas of the latter curves, taken from x = 0 to x — TT,
are the values of the coefficients a, 6, c, d, &c., in the equation

I

~ TT <f> (x) = a sin x + b sin 2a? + c sin 3x + d sin 4# + &c.

221. We can verify the foregoing equation (D), (Art. 220),
by determining directly the quantities alt «2, a3, ... a.y &c., in the
equation

<£ (a?) = a: sin a? + a2 sin 2# + a3 sin 3x + . . . a, sin Jic + &e. ;

for this purpose, we multiply each member of the latter equation
by sin ix dx, i being an integer, and take the integral from x = 0
to X = TT, whence we have

I <f)(x) sin ix dx = ax I sin x sin ix dx + «2 (sin 2# sm'ix dx + &c.

+ aj I sinjx sin ix dx + ... &c.

Now it can easily be proved, 1st, that all the integrals, which
enter into the second member, have a nul value, except only the

term aL \ sin ix sin ixdx ; 2nd, that the value of Ismixsmixdx is
i-TT ; whence we derive the value of ai} namely

2 r

- I (f> (a?) sin ix dx.

The whole problem is reduced to considering the value of the
integrals which enter into the second member, and to demon- -i
strating the two preceding propositions. The integral

2 I sin jjc si 11 ixdx,

JL

188 THEORY OF HEAT. [CHAP. III.

taken from x = 0 to x — TT, in which i and j are integers, is
j—j sin (* - j) x - ^-. sin (i + j) x + C.

Since the integral must begin when x = 0 the constant C is
nothing, and the numbers i and j being integers, the value of the
integral will become nothing when OJ = TT; it follows that each
of the terms, such as

at \ sin x sin ix da, a2 1 sin 2x sin ix doc, a3 (sin 5x sin ixdxt &c.,

vanishes, and that this will occur as often as the numbers i and j
are different. The same is not the case when the numbers i and j

are equal, for the term - — .sin (i —j) x to which the integral re-

j

duces, becomes -^ , and its value is TT. Consequently we have

0

2 I sin ix sin ix dx == TT ;

thus we obtain, in a very brief manner, the values of alt az, a3) ...
«4, &c., namely,

2 f 2 f

ttj = - /(£ (#) sin # dr, a2 = - l<£ (x) sin 2

2 f 2 r

#3 = - I c/> (a?) sin 3# &e, a, = - \$(x) sin 10

Substituting these we have

%7r(f> (x) = sin x I </> (a?) sin # cZic + sin 2x l(f) (x) sin 2# J^? + &c.

+ sin ix 1 0 (a?) sin ixdx + &c.

222. The simplest case is that in which the given function
has a constant value for all values of the variable x included

between 0 and TT ; in this case the integral I sin ixdx is equal to

9

?, if the number i is odd, and equal to 0 if the number i is even.

SECT. VI.] LIMITS OF THE DEVELOPMENTS. 180

Hence we deduce the equation

.j TT = sin x + g sin 3# 4- - sin 5# -f = sin 7x + &c., (N t

which has been found before.

It must be remarked that when a function <f> (x) has been de
veloped in a series of sines of multiple arcs, the value of the series

a sin x -f & sin 2# + c sin 3x + d sin kx + &c.

is the same as that of the function $ (#) so long as the variable x
is included between 0 and IT ; but this equality ceases in general
to hold good when the value of x exceeds the number TT. ~

Suppose the function whose development is required to be x,
we shall have, by the preceding theorem, •

2 irx = sin x I x sin x dx + sin 2x I x sin 2# dx

+ sin 3# I x sin 3# dx 4- &c.

The integral I x sin i#cfa? is equal to f T ; the indices 0 and TT,
/« z

which are connected with the sign I , shew the limits of the inte

gral ; the sign -f must be chosen when i is odd, and the sign —
when i is even. We have then the following equation,

•^x = sin x — = sin 2# + ^ sin 3# — -j sin 4# + - sin 5^ — &c.
— 25 v 4 o

223. We can develope also in a series of sines of multiple
arcs functions different from those in which only odd powers of
the variable enter. To instance by an example which leaves no
doubt as to the possibility of this development, we select the
function cos x, which contains only even powers of xt and which it \
may be developed under the following form :

a sin x + 6 sin 2x + c sin 3# + d sin 4<x + e sin 5# + &c.,

*r

although in this series only odd powers of the variable enter.

190 THEORY OF HEAT. [CHAP. III.

We have, in fact, by the preceding theorem,
- TT cos x — sin x I cos x sin x dx + sin 2# I cos x sin 2# dx

4- sin 3x I cos x sin 3# cfce + &c.
The integral I cos x sin ix dx is equal to zero when i is an

odd number, and to .2_\ when i is an even number. Supposing
successively i = 2, 4, 6, 8, etc., we have the always convergent

seres

TTT cos x = =— s sm 2# + ^— ? sin 4^ + K— "7 sin
4 I . o o . o o . /

or,

This result is remarkable in this respect, that it exhibits the
development of the cosine in a series of functions, each one of
which contains only odd powers. If in the preceding equation x
be made equal to JTT, we find

This series is known (Introd. ad analysin. infiniL cap. x.).

224. A similar analysis may be employed for the development
of any function whatever in a series of cosines of multiple arcs.

Let <£(#) be the function whose development is required, we
may write

<£ (x) — aQ cos Ox + at cos x + aa cos Zx + aa cos 3x + &c.

+ aicosix+&c ........... (m).

If the two members of this equation be multiplied by cosjx,
and each of the terms of the second member integrated from
x = 0 to x = TT ; it is easily seen that the value of the integral
will be nothing, save only for the term which already contains
cosjx. This remark gives immediately the coefficient a,; it is
sufficient in general to consider the value of the integral

Icoajx cos ix dx,

SECT. VI.] DEVELOPMENT IN SERIES OF COSINES. 191

taken from x = 0 to x — IT, supposing j and i to be integers. We
have

This integral, taken from x = 0 to x — TT, evidently vanishes
whenever j and i are two different numbers. The same is not
the case when the two numbers are equal. The last term

sn -

becomes ~ ,| and its value is \TT, when the arc x is equal to 77%

If then we multiply the two terms of the preceding equation (m)
by cos ix, and integrate it from 0 to TT, we have

</> (X) COS IX dx = ^TTdi,

an equation which exhibits the value of the coefficient c^.

To find the first coefficient «0, it may be remarked that in
the integral

i t

dn (j—i) x,

if j = 0 and i = 0 each of the terms becomes ^ , and the value

of each term is JTT ; thus the integral I cos jx cos ix dx taken

from x = 0 to x = TT is nothing when the two integers j and i
are different : it is \tr when the two numbers j and i are equal
but different from zero ; it is equal to TT when j and i are each
equal to zero ; thus we obtain the following equation,

1 fv [" fir

2 Jo Jo Jo

+ cos 3# I </> (a?) cos 3# d# + &c. (n)\

J o

1 The process analogous to (A) in Art. 222 fails here ; yet we see, Art. 177, that
an analogous result exists. [B. L. E.]

192 THEORY OF HEAT. [CHAP. III.

This and the preceding theorem suit all possible functions,
whether their character can be expressed by known methods of
analysis, or whether they correspond to curves traced arbitrarily.

225. If the proposed function whose development is required
in cosines of multiple arcs is the variable x itself ; we may write
down the equation

1

» TTX = a0 + ttj cos x + a2 cos Zx -f a3 cos ox+ ... + at cos ix + &c.,

and we have, to determine any coefficient whatever ait the equa
tion at = I x cos ix dx. This integral has a nul value when i is

'o

2
an even number, and is equal to — -^ when i is odd. We have at

the same time a0 = 7 ?r2. We thus form the following series,

1 A cos x . cos 3# , cos 5% . cos 7x 0

x = ~ TT — 4 4 — ^ 4— ^3— — 4 -^ &c.

2 7T d 7T O7T / 7T

We may here remark that we have arrived at three different
developments for x, namely,

1 1111

- x — sin x — ^ sin 2x + - sin 3# — -r sin ^x + - sin 5x — &c.,
— « tj jb o

12. 2 2

- x = - sin oj — ^— sin 3^ + r^— sin 5^c - &c. (Art. 181),

2 TT 3V 5V

112 2 2

^X = jTT COSOJ — ^— COS <$X — -^— COS 5x — &C.

2 4 TT 3V 5V

It must be remarked that these three values of \x ought not
to be considered as equal; with reference to all possible values of
x, the three preceding developments have a common value only
when the variable x is included between 0 and JTT. The con
struction of the values of these three series, and the comparison of
the lines whose ordinates are expressed by them, render sensible
the alternate coincidence and divergence of values of these
functions.

To give a second example of the development of a function in
a series of cosines of multiple arcs, we choose the function sin a?,

SECT. VI.] TRIGONOMETRICAL DEVELOPMENTS. 193

which contains only odd powers of the variable, and we may sup
pose it to be developed in the form

a -j- b cos x -f c cos 2x + d cos Sx -f &c.

Applying the general equation to this particular case, we find,
as the equation required,

1 . 1 cos 2# cos 4# cos

_ __.._..__..__

0
_&<..

Thus we arrive at the development of a function which con
tains only odd powers in a series of cosines in which only even
powers of the variable enter. If we give to a? the particular value
JTT, we find

111111

57r==2+rjr375 + o\7-fT9 +

Now, from the known equation,

we derive
1

and also

1111

^ 7T = — —

-&c.

2 3.5 7.9 11.13
Adding these two results we have, as above,
111111 1

T 7T = 7^ + ^ — ^ — 0 "—^ + ~ — -^ — ^ — pr + TT — r^ — &C.

4 2 1.3 3.o o.7 7.9 9.11

226. The foregoing analysis giving the means of developing
any function whatever in a series of sines or cosines of multiple
arcs, we can easily apply it to the case in which the function to be
developed has definite values when the variable is included
between certain limits and has real values, or when the variable is
included between other limits. We stop to examine this particular
case, since it is presented in physical questions which depend on
partial differential equations, and was proposed formerly as an ex
ample of functions which cannot be developed in sines or cosines
F. H. 13

THEORY OF HEAT. [CHAP. III.

of multiple arcs. Suppose then that we have reduced to a series of
this form a function whose value is constant, when x is included
between 0 and a, and all of whose values are nul when x is in
cluded between a and IT. We shall employ the general equation
(D}y in which the integrals must be taken from x = 0 to x = TT.
The values of <£(.x) which enter under the integral sign being
nothing from x = a to x = TT, it is sufficient to integrate from x — 0
to x = a. This done, we find, for the series required, denoting by
h the constant value of the function,

1 f l-cos2a

~7r<£(#) = h <(I — cos a) sm x -\ -- ~ - sin 2x

1 — cos 3a .

_j -- -- sm ^x + &C.
o

If we make /t = JTT, and represent the versed sine of the arc x
by versin x, we have

<£ (x] = versin a sin a; + ^ versin 2a sin 2# + ^ versin 3 a sin 3# + &C.1

This series, always convergent, is such that if we give to x any
value whatever included between 0 and a, the sum of its terms
will be ^TT ; but if we give to x any value whatever greater than
a and less than 4?r, the sum of the terms will be nothing.

In the following example, which is not less remarkable, the
values of $ (x} are equal to sin — - for all values of x included

between 0 and a, and nul for values of as between a and TT. To
find what series satisfies this condition, we shall employ equa
tion (Z>).

The integrals must be taken from x = 0 to x = IT ; but it is
sufficient, in the case in question, to take these integrals from
x = 0 to x = a, since the values of <f> (x) are supposed nul in the
rest of the interval. Hence we find

sin as sin 2a sin Zx sin 3a sin 3x „

+ ~ + ~- + &c

1 In what cases a function, arbitrary between certain limits, can be developed
in a series of cosines, and in what cases in a series of sines, has been shewn by
Sir W. Thomson, Cainb. Math. Journal, Vol. n. pp. 258—262, in an article
signed P. Q. K., On Fourier's Expansions of Functions in Trigonometrical Series.

SECT. VI.] TRIGONOMETRICAL DEVELOPMENTS. 195

If a be supposed equal to TT, all the terms of the series vanish,
except the first, which becomes - , and whose value is sin x • we
have then <#

227. The same analysis could be extended to the case in
which the ordinate represented by $(x) was that of a line com
posed of different parts, some of which might be arcs of curves
and others straight lines. For example, let the value of the func
tion, whose development is required in a series of cosines of

multiple arcs, be \^\ -a?} from x = 0 to x = JTT, and be nothing

from x = JTT to x = TT. We shall employ the general equation (n), - /*
and effecting the integrations within the given limits, we find "

that the general term1 I U^J - x 2 cos ixdx is equal to/-3 when i
is even) to 4- ^ when i is the double of an odd number, and to

?,

— -^ when i is four times an odd number. On the other hand, we

-I 3 -. ,.

3 •? for the value of tte first term 9 fa&y&e. We have then

the following development :

2 cosa; cos %x cos oas cos

< =

cos 2ic cos 4# cos 6#

~J 2^ 42 ~* ^2 &c-

The second member is represented b}' a line composed of para
bolic arcs and straight lines.

228. In the same manner we can find the development of a
function of x which expresses the ordinate of the contour of a
trapezium. Suppose <f>(x) to be equal to x from x = 0 to x = a,
that the function is equal to a from x — a. to x — IT — a, and lastly
equal to TT - x, from x = TT - a to x = IT. To reduce it to a series

? * ^ * *^ tf*l»>

^n )',£, 13—2

196 THEORY OF HEAT. [CHAP. III.

of sines of multiple arcs, we employ the general equation (D).
The general term /<£ (x) sin ix dx is composed of three different

2
parts, and we have, after the reductions, -^sin ia for the coefficient

of sin ix, when i is an odd number ; but the coefficient vanishes
when i is an even number. Thus we arrive at the equation

-7T(j)(x) = 2\ sin a. sin x + ^ sin 3a sin 3# 4- —^ sin 5a sin 5x

Zi (^ O O

+ 5=2 sin 7a sin 7# 4- &c. [• (X).1

If we supposed a = JTT, the trapezium would coincide with an
isosceles triangle, and we should have, as above, for the equa
tion of the contour of this triangle,

~ 7r<f> (as) = 2 (sin a? — ^ sin 3# + ^ sin 5% — ^ sin 7# + &c. k2
2 \ d 0 / j

a series which is always convergent whatever be the value of x.
In general, the trigonometric series at which we have arrived,
in developing different functions are always convergent, but it
has not appeared to us necessary to demonstrate this here ; for the
terms which compose these series are only the coefficients of terms
of series which give the values of the temperature ; and these
coefficients are affected by certain exponential quantities which
decrease very rapidly, so that the final series are very convergent.
With regard to those in which only the sines and cosines of
multiple arcs enter, it is equally easy to prove that they are
convergent, although they represent the ordinates of discontinuous
lines. This does not result solely from the fact that the values
of the terms diminish continually ; for this condition is not
sufficient to establish the convergence of a series. It is necessary
that the values at which we arrive on increasing continually the
number of terms, should approach more and more a fixed limit,

1 The accuracy of this and other series given by Fourier is maintained by
Sir W. Thomson in the article quoted in the note, p. 194.

2 Expressed in cosines between the limits 0 and TT,

ITT<P («)=__{ cos.2a; + - cos Gx + ^- cos Wx + &c. ) .

o \ O O /

Cf. De Morgan's Diff. and Int. Calc., p. 622. [A. F.]

SECT. VI.] GEOMETRICAL ILLUSTRATION. 197

and should differ from it only by a quantity which becomes less
than any given magnitude: this limit is the value of the series.
Now we may prove rigorously that the series in question satisfy
the last condition.

229. Take the preceding equation (X) in which we can give
to x any value whatever; we shall consider this quantity as a
new ordinate, which gives rise to the following construction.

Having traced on the plane of x and y (see fig. 8) a rectangle
whose base OTT is equal to the semi-circumference, and whose
height is £?r ; on the middle point m of the side parallel to the
base, let us raise perpendicularly to the plane of the rectangle
a line equal to |TT, and from the upper end of this line draw
straight lines to the four corners of the rectangle. Thus will be
formed a quadrangular pyramid. If we now measure from the
point 0 on the shorter side of the rectangle, any line equal to a,
and through the end of this line draw a plane parallel to the base
OTT, and perpendicular to the plane of the rectangle, the section
common to this plane and to the solid will be the trapezium whose
height is equal to a. The variable ordinate of the contour of
this trapezium is equal, as we have just seen, to

^ sm 3a sm %x + 7z sm ^a sm

O O

— (sin a sin x

7T \

It follows from this that calling x, y, z the co-ordinates of any
point whatever of the upper surface of the quadrangular pyramid
which we have formed, we have for the equation of the surface
of the polyhedron, between the limits

1 sin x sin y sin 3x sin 3^ sin 5x sin oy

-TTZ = -- j2— 32 - ^2 - ^-

198 THEORY OF HEAT. [CHAP. III.

This convergent series gives always the value of the ordinate
z or the distance of any point whatever of the surface from the
plane of x and y.

The series formed of sines or cosines of multiple arcs are
therefore adapted to represent, between definite limits, all possible
functions, and the ordinates of lines or surfaces whose form is
discontinuous. Not only has the possibility of these develop
ments been demonstrated, but it is easy to calculate the terms
of the series; the value of any coefficient whatever in the
equation

<j) (x) = ax sin x -f <32 sin 2# + a3 sin 3# + . . . -f at sin ix + etc.,
is that of a definite integral, namely,

2

- \d> (as) sin i
TT J

ix dx.

Whatever be the function <£ (x), or the form of the curve
which it represents, the integral has a definite value which may
be introduced into the formula. The values of these definite

integrals are analogous to that of the whole area I (/> (x) dx in
cluded between the curve and the axis in a given interval, or to
the values of mechanical quantities, such as the ordinates of the
centre of gravity of this area or of any solid whatever. It is
evident that all these quantities have assignable values, whether
the figure of the bodies be regular, or whether we give to them
an entirely arbitrary form.

230. If we apply these principles to the problem of the motion
of vibrating strings, we can solve difficulties which first appeared
in the researches of Daniel Bernoulli. The solution given by this
geometrician assumes that any function whatever may always be
developed in a series of sines or cosines of multiple arcs. Now
the most complete of all the proofs of this proposition is that
which consists in actually resolving a given function into such a
series with determined coefficients.

In researches to which partial differential equations are ap
plied, it is often easy to find solutions whose sum composes a
more general integral ; but the employment of these integrals
requires us to determine their extent, and to be able to dis-

SECT. VI.] REMARKS ON THE DEVELOPMENTS. 199

tinguish clearly the cases in which they represent the general
integral from those in which they include only a part. It is
necessary above all to assign the values of the constants, and
the difficulty of the application consists in the discovery of the
coefficients. J^is remarkable that we can express by convergent
series, and, as we shalPsee Tn the sequel, by definite integrals,
the ordinates of lines and surfaces which arenot subject to a
_ continuous law1. We see by this that we must admit into analysis
functionswKich have equal values, whenever the variable receives
any values whatever included between two given limits, even
though on substituting in these two functions, instead of the
variable, a number included in another interval, the results of
the two substitutions are not the same. The functions which
enjoy this property are represented by different lines, which
coincide in a definite portion only of their course, and offer a
singular species of finite osculation. These considerations arise
in the calculus of partial differential equations; they throw a new
light on this calculus, and serve to facilitate its employment in
physical theories.

231. The two general equations which express the develop
ment of any function whatever, in cosines or sines of multiple
arcs, give rise to several remarks which explain the true meaning
of these theorems, and direct the application of them.

If in the series

a + b cos x + c cos 2x + d cos 3# + e cos 4>x + &c.,

we make the value of x negative, the series remains the same ; it t ^
also preserves its value if we augment the variable by any multiple
whatever of the circumference 2?r. Thus in the equation

- TT<£ (x) = x I </> (x) dx -f cos x l(f> (x) cos xdx

+ cos 2# Iff) (x) cos 2xdx + cos 3# /</> (x) cos Sxdx + &c....(i/),

the function $ is periodic, and is represented by a curve composed
of a multitude of equal arcs, each of which corresponds to an

1 Demonstrations have been supplied by Poisson, Deflers, Dirichlet, Dirksen,
Bessel, Hamilton, Boole, De Morgan, Stokes. See note, pp. 208, 209. [A. F.]

200

THEORY OF HEAT.

[CHAP. IJI.

interval equal to STT on the axis of the abscissae. Further, each of
these arcs is composed of two symmetrical branches, which cor
respond to the halves of the interval equal to 2?r,

Suppose then that we trace a line of any form whatever </><£a
(see fig. 9.), which corresponds to an interval equal to TT.

Fig. 9.

If a series be required of the form

a + b cos x + c cos 2% + d cos 3x -f &c.,

such that, substituting for x any value whatever X included be
tween 0 and TT, we find for the value of the series that of the
ordinate X<j>, it is easy to solve the problem : for the coefficients
given by the equation (v) are

if 2

- l<f>(x) dx, -

2 r

, - l(f> (x) cos §xdxt &c.

These integrals, which are taken from x = 0 to x — TT, having
always measurable values like that of the area Ofon, and the
series formed by these coefficients being always convergent, there
is no form of the line <£</>a, for which the ordinate X(j> is not
exactly represented by the development

a -f "b cos x -\- c cos 2# + d cos 3# -f e cos

&c.

The arc <£(/>a is entirely arbitrary ; but the same is not the case
with other parts of the line, they are, on the contrary, determinate;
thus the arc <£a which corresponds to the interval from 0 to — TT is
the same as the arc </>a ; and the whole arc a<pa is repeated on
consecutive parts of the axis, whose length is 2?r.

We may vary the limits of the integrals in equation (v). If
they are taken from x = — ?r to x = TT the result will be doubled :
it would also be doubled if the limits of the integrals were
0 and 27rr instead of being 0 and TT. We denote in general by the

SECT. VI.]

GEOMETRICAL ILLUSTRATION.

201

i i

7T(f) (x) = ^ <j> (x] dx + cos x

ft

sign I an integral which begins when the variable is equal to a,

J a

and is completed when the variable is equal to b ; and we write
equation (n) under the following form :

r*

(x) cos x dx -f cos 2x (f> (x} cos 2xdx

Jo

[n

+ cos 3x $ (x) cos %xdx + etc ........... (V).

J 0

Instead of taking the integrals from x = 0 to x — TT, we might
take them from x = 0 to x = 2?r, or from x — — IT to x = TT; but in
each of these two cases, TT</> (x} must be written instead of JTT^ (a:)
in the first member of the equation.

232. In the equation which gives the development of any
function whatever in sines of multiple arcs, the series changes
sign and retains the same absolute value when the variable x
becomes negative; it retains its value and its sign when the
ariable is increased or diminished by any multiple whatever of /

Fig. 10.

v

the circumference 2?r. The are ^a (see fig. 10), which cor
responds to the interval from 0 to TT is arbitrary; all the other
parts of the line are determinate. The arc </>(£a, which corresponds
to the interval from 0 to — TT, has the same form as the given arc
(fxfra ; but it is in the opposite position. The whole arc OLffxjxfxjxi is
repeated in the interval from TT to 3?r, and in all similar intervals.
We write this equation as follows :

- TT<£ (a;) = sin x I (f> (x) sin xdx + sin 2x I <f> (x) sin Zxdx
2 Jo Jo

+ sin 3x I (j> (x) sin 3xdx + &c.

202 THEORY OF HEAT. [CHAP. III.

We might change the limits of the integrals and write

/•2/r T+T r-n

I or I instead of I ;

J 0 J _7T JO

but in each of these two cases it would be necessary to substitute
in the first member TT<£ (x) for JTT<£ (x).

233. The function <£ (x) developed in cosines of multiple arcs,
is represented by a line formed of two equal arcs placed sym-

Fig. 11.

metrically on each side of the axis of y, in the interval from
— TT to +TT (see fig. 11) ; this condition is expressed thus,

The line which represents the function i|r (x) is, on the contrary,
formed in the same interval of two opposed arcs, which is what is
expressed by the equation

Any function whatever F(x\ represented by a line traced
arbitrarily inTEe interval from — TT to + TT, may always be divided
into two functions such as <£ (V) and ^H[g)« In fact, if the line
F'F'mFF represents the function F(x}} and we raise at the point
o the ordinate om, we can draw through the point m to the right
of the axis om the arc mff similar to the arc mF'F of the given
curve, and to the left of the same axis we may trace the arc mff'
similar to the arc mFF '; we must then draw through the point m
a line <^<^'m^ which shall divide into two equal parts the differ
ence between each ordinate ooF or x ' f and the corresponding

SECT. VI.] GEOMETRICAL DEMONSTRATION. 203

ordinate of or x'F'. We must draw also the line -vJ/^'ChJ^ whose
ordinate measures the half-difference between the ordinate of
F'F'mFF and that of f'f'mff. This done the ordinate of the
lines FF'mFF, and f'f'mff being denoted by F (x) and f(x)
respectively, we evidently have /(a?) = F(— x) ; denoting also the
ordinate of $'$m$$ by <£ (x), and that of iJrSJr'Oi/nJr by ^ (x),
we have

F(x) = <j, (x) + f (x) and f(x) = $(x}-^(x}=F (- x),
hence

<£ (x) = i*» + lF(- x) and + (*) = §*» - ^(-*),

whence we conclude that

<$>(x) = $(-x) and ^ (x) = - ^ (- a?),
which the construction makes otherwise evident.

Thus the two functions (/> (x) and i|r (x), whose sum is equal to
F (at) may be developed, one in cosines of multiple arcs, and the
other in sines.

If to the first function we apply equation (v), and to the second
the equation (/x), taking the integrals in each case from x = - TT
to X = TT, and adding the two results, we have

2 /£(*) ^ + cos x |^{*) cos ^ ^ + cos 2a? /</) (a;) cos 2% dx + &c.

+ sin x^r(x} sin re dx + sin 2# ^(#) sin 2aj Ja; + &c.

The integrals must be taken from x = — TT to x = IT. It may now

f+7r
be remarked, that in the integral I <£ (x) cos a? cfo we could,

J -IT

without changing its value, write 0 (x) + -^ (a?) instead of <£> (a?) :
for the function cos a? being composed, to right and left of the

axis of xt of two similar parts, and the function ^r (x) being, on the

r+Tr
contrary, formed of two opposite parts, the integral I ty(x) cos xdx

J -IT

vanishes. The same would be the case if we wrote cos 2a; or
cos 3a-, and in general cos ix instead of cos a?, i being any integer

204 THEORY OF HEAT. [CHAP. III.

r+7T

from 0 to infinity. Thus the integral I <£ (x) cos ix dx is the same

J -77

as the integral

r+ir r+n

I bfr (%) + ^ (X)J cos ™ dx, or I F(x] cos ix dx.

J "IT J -IT

r+T
It is evident also that the integral I ^(x) smixdx is equal

J -TT

/*+«• /*+"•

to the integral I F(x] sin ixdx, since the integral I </>(#) swi

J -7T J -TT

vanishes. Thus we obtain the following equation (p), which serves
to develope any function whatever in a series formed of sines and
cosines of multiple arcs :

cos x

\ F[x] cos x dx + cos 2# I F(x] cos 2x dx + &c.
+ sin x \ F(x] sin x dx + sin 2x I F(x) sin 2x dx + &c.

234. The function F(x), which enters into this equation, is
represented by a line F'F'FF, of any form whatever. The arc
F'F'FF, which corresponds to the interval from.— TT to +TT, is
arbitrary ; all the other parts of the line are determinate, and the
arc F'F'FF is repeated in each consecutive interval whose length
is 27T. We shall make frequent applications of this theorem, and
of the preceding equations (ft) and (i/).

If it be supposed that the function F(x] in equation (p) is re
presented, in the interval from — IT to + TT, by a line composed of
two equal arcs symmetrically placed, all the terms which contain
sines vanish, and we find equation (v). If, on the contrary, the
line which represents the given function F(x) is formed of two
equal arcs opposed in position, all the terms which do not contain
sines disappear, and we find equation (/x). Submitting the func
tion F(x) to other conditions, we find other results.

If in the general equation (p) we write, instead of the variable
x, the quantity -— , x denoting another variable, and 2r the length

SECT. VI.] MODIFICATION OF THE SERIES. 205

of the interval which includes the arc which represents F(x}\
the function becomes F (— j, which we may denote by /(#).

The limits x = — TT and x = TT become — = — TT. — = TT ; we

r r

have therefore, after the substitution,

<p?

X [ „ . 7T# , 27T.T f ,, . 277-tf ,

-f cos TT - I f(x) cos — dx-\- cos I / (x) cos c&e f etc.

x f ., N . TTX j . 27r# /* ,. . . 2?nr ,
+ sin TT - I /(a?) sin — dx -f sm — — \f(x) sm d# + etc.

All the integrals must be taken like the first from x = — r to
x = +r. If the same substitution be made in the equations (v)
and (yu,), we have

cos -- I f(x) cos — dx

2?ra; /*/./» 27ra;
+ cos- - \f(x) cos --

1 /., x . 7T5? F ~f

2 r/W = sm — \ f(x

^ ' J 0

In the first equation (P) the integrals might be taken from
from x = 0 to x = 2r, and representing by x the whole interval 2r,
we should have *

1 It has been shewn by Mr J. O'Kinealy that if the values of the arbitrary
f unction /(x) be imagined to recur for every range of x over successive intervals X,
we have the symbolical equation

and the roots of the auxiliary equation being

t»— ^J— ^— , 7« = 0, 1, 2, 3... cc, [Turn over.

206 THEORY OF HEAT. [CHAP. III.

x}dx (II)

27T03 f-, . ZTTX , 4-TnE f .. , 47nc , ,

-f cos -yr- I / (x) cos -TF- a# + cos — =^— I /(a?) cos -^- a# 4- &c.

. ZTTX f /. / N . 27HB 7 . 4urx [ - , . . 47r# , p
-f sin — TT- / (a?) sin — TT- a# + sin - v Ifw sin -^- a^ + &c.
JL J .A- -A- J •&

235. It follows from that which has been proved in this sec
tion, concerning the development of functions in trigonometrical
series, that if a function f(x) be proposed, whose value in a de
finite interval from x = 0 to x = X is represented by the ordinate
of a curved line arbitrarily drawn ; we can always develope this
function in a series which contains only sines or only cosines, or
the sines and cosines of multiple arcs, or the cosines only of odd
multiples. To ascertain the terms of these series we must employ
equations (M), (N), (P).

The fundamental problems of the theory of heat cannot be
completely solved, without reducing to this form the functions
which represent the initial state of the temperatures.

These trigonometric series, arranged according to cosines or
sines of multiples of arcs, belong to elementary analysis, like the
series whose terms contain the successive powers of the variable.
The coefficients of the trigonometric series are definite areas, and
those of the series of powers are functions given by differentiation,
in which, moreover, we assign to the variable a definite value. We
could have added several remarks concerning the use and pro
perties of trigonometrical series ; but we shall limit ourselves to
enunciating briefly those which have the most direct relation to
the theory with which we are concerned.

it follows that

f(x) =A0 + AI cos — — 1- ^2 cos 2 — — h^3cos 3 — — + &c.

A A A

. 2-7TX _ STTX ITTX

+ B^ sin -r— + B2 sin 2 — - + B% sin 3 -r— + &c.
A A A

The coefficients being determined in Fourier's manner by multiplying both

sides by . n —. — - and integrating from 0 to X. (Philosophical Magazine, August
sin A

1874, pp. 95, 9G). [A. F.j

SECT. VI.] REMARKS ON THE SERIES. 207

1st. The series arranged according to sines or cosines of mul
tiple arcs are always convergent ; that is to say, on giving to the
variable any value whatever that is not imaginary, the sum of the
terms converges more and more to a single fixed limit, which is
the value of the developed function.

2nd. If we have the expression of a function f(x) which cor
responds to a given series

a + b cos x + c cos 2x + d cos 3# + e cos 4# + &c.,
and that of another function </> (a?), whose given development is

Q.+ ft cos x + 7 cos Zx + 8 cos 3x + e cos 4?x -f &c.,
it is easy to find in real terms the sum of the compound series

act + b/3 + cy -f dS + ee + &C.,1
and more generally that of the series

ax + 6/3 cos x + cy cos 2# + cZS cos 3# + ee cos 4tx + &c.,
which is formed by comparing term by term the two given series.
This remark applies to any number of series.

3rd. The series (5^) (Art. 234s) which gives the development
of a function F (x) in a series of sines and cosines of multiple arcs,
may be arranged under the form

+ cos x \ F(a) cos ado. + cos 2# I F (a) cos 2s>cZa -f &c.
+ sin x I F (a) sin acZa + sin 2x I F (a) sin 2a dx + &c.

a being a new variable which disappears after the integrations.
We have then

+ cos x cos a + cos 2x cos 2a + cos 3# cos 3a + &c.
+ sin cc sin a + sin 2x sin 2a + sin Sx sin 3a + &c. ,

1 We shall have

fir

f

Jo

t(x)<f>(x)dx=CMT+lT{bp+Cy+...}. [R. L. E.]

208 THEORY OF HEAT. [CHAP. III.

or

F(x) = - I F(-J) doi Ji + cos (x - a) + cos 2 (x - a) + &c. j .

Hence, denoting the sum of the preceding series by

2 cos i (x — a)
taken from i = 1 to i = GO , we have

F(x)=- \F (a) d* \l + S cos i(x - a)! .

7TJ [Z J

The expression ^ + X cos i (a? — a) represents a function of #

2

and a, such that if it be multiplied by any function whatever F(oi),
and integrated with respect to a between the limits a = — TT and
a = ?r, the proposed function jP(a) becomes changed into a like
function of x multiplied by the semi-circumference TT. It will be
seen in the sequel what is the nature of the quantities, such as

5 + 2cos*(# — a), which enjoy the property we have just enun-

2

ciated.

4th. If in the equations (M), (N), and (P) (Art 234), which
on being divided by r give the development of a function f(x),
we suppose the interval r to become infinitely large, each term of
the series is an infinitely smal^ element of an integral; the sum of
the series is then represented by a definite integral. When the
bodies have determinate dimensions, the arbitrary functions which
represent the initial temperatures, and which enter into the in
tegrals of the partial differential equations, ought to be developed
in series analogous to those of the equations (M), (N), (P) ; but
\ these functions take the form of definite integrals, when the
dimensTons 'of the bodies are not determinate, as will be ex
plained in the course of this work, in treating of the free diffusion
of heat (Chapter IX.).

Note on Section VI. On the subject of the development of a function whose
values are arbitrarily assigned between certain limits, in series of sines and
cosines of multiple arcs, and on questions connected with the values of such
series at the limits, on the convergency of the series, and on the discontinuity
of their values, the principal authorities are

Poisson. Theorie mathematiqiie de la Chaleur, Paris, 1835, Chap. vn. Arts.
92 — 102, Sur la maniere d'exprimcr les fonctions arbitraircs par des series de

SECT. VII.] LITERATURE. 209

quantites periodiqucs. Or, more briefly, in his TraiU de Mecanique, Arts. 325—328.
Poisson's original memoirs on the subject were published in the Journal de VEcole
Poll/technique, Cahier 18, pp. 417 — 489, year 1820, and Cahier 19, pp. 404—509,
year 1823.

De Morgan, Differential and Integral Calculus. London, 1842, pp. 609 — 617.
The proofs of the developments appear to be original. In the verification of the
developments the author follows Poisson's methods.

Stokes, Cambridge Philosophical Transactions, 1847, Vol. VIH. pp. 533 — 556.
On the Critical i-alucs of the sums of Periodic Series. Section I. Mode of ascertain
ing the nature of the discontinuity of a function which is expanded in a series
of sines or cosines, and of obtaining the developments of the derived functions.
Graphically illustrated.

Thomson and Tait, Natural Philosophy, Oxford, 1867, Vol. I. Arts. 75—77.
Donkin, Acoustics, Oxford, 1870, Arts. 72 — 79, and Appendix to Chap. rv.
Matthieu, Cours de Physique Mathematique, Paris, 1873, pp. 33 — 36.
Entirely different methods of discussion, not involving the introduction of
arbitrary multipliers to the successive terms of the series were originated by

Dirichlet, Crelle's Journal, Berlin, 1829, Band iv. pp. 157 — 169. Sur la con
vergence des series trigonome'triques qui servent a rcpresenter une fonction arbitraire
entre les limites donnees. The methods of this memoir thoroughly deserve at
tentive study, but are not yet to be found in English text-books. Another memoir,
of greater length, by the same author appeared in Dove's Repertorium der Phyaik,
Berlin, 1837, Band i. pp. 152 — 174. Ueber. die Darstellung ganz willkuhrlicher
Functioncn durch Sinus- und Cosinusreihen. Von G. Lejeune Dirichlet.
Other methods are given by

Dirksen, Crelle's Journal, 1829, Band iv. pp. 170^178. Ueber die Convergenz
einer nach den Sinussen imd Cosinussen der Vielfachen eines Winkel* fortachreit en-
den Eeihe.

Bessel, Astronomische Nachrichten, Altona, 1839, pp. 230 — 238. Ueber den

Amdruck einer Function $ (x) durch Cosinusse und Sinusse der Vielfachen von x.

The writings of the last three authors are criticised by Biemann, Gesammelte

Mathematische Werke, Leipzig, 1876, pp. 221 — 225. Ueber die Darstellbarkeit einer

Function durch eine Trigonometrische Eeihe.

On Fluctuating Functions and their properties, a memoir was published by
Sir W. K. Hamilton, Transactions of the Royal Irish Academy, 1843, Vol. xix. pp.
264 — 321. The introductory and concluding remarks may at this stage be studied.
The writings of Deflers, Boole, and others, on the subject of the expansion
of an arbitrary function by means of a double integral (Fourier's Theorem) will
be alluded to in the notes on Chap. IX. Arts. 361, 362. [A. F.]

SECTION VII.

Application to the actual problem.

236. We can now solve in a general manner the problem of
the propagation of heat in a rectangular plate BAG, whose end A
is constantly heated, whilst its two infinite edges B and C are
maintained at the temperature 0.
F. H.

210 THEORY OF HEAT. [CHAP. ITT.

Suppose the initial temperature at all points of the slab BAG
to be nothing, but that the temperature at each point in of the
edge A is preserved by some external cause, and that its fixed
value is a function f(x) of the distance of the point m from the
end 0 of the edge A whose whole length is 2r; let v be the
constant temperature of the point m whose co-ordinates are x and
y, it is required to determine v as a function of x and y.

The value v = ae~mv sin mx satisfies the equation

HT

a and m being any quantities whatever. If we take m = i - ,

i being an integer, the value ae *" r sin — • vanishes, when x = r,

whatever the value of y may be. We shall therefore assume, as a
more general value of v,

. - . - . „

v = a,e r sin -- \- at>e r sin - + a»e r sin -- h &c.
r r r

If y be supposed nothing, the value of v will by hypothesis
be equal to the known function f(x). We then have

/., x . . . 0

j (x) = a^ sin — + &2 sin -- \- aa sin -- f- &c.

The coefficients alt a2, «3, &c. can be determined by means of
equation (M), and on substituting them in the value of v we have

1 -IT- . TTX ^ ,, N . 7TX , -2ir^ . %irX C -, , . 27T# 7

s rv = e r sm — / /(a?) sm — a^ + e f sin - f (») sm -- dx

2 r-7 r r J \ i T

o" . V'IM/ I /»/ A * «*««V j| .0

+ e r sin / f (x) sin dx + &c.

237. Assuming r = TT in the preceding equation, we have the
solution under a more simple form, namely

-'jrv — e^ sin x\f(x] sin #<£& + e~2y sin 2# !./(#) sin Zxdx

+ e~5v sin 3^7 / f(x\ sin 3a?c?^ + &c /. (a

J-7 w

SECT. VII.] APPLICATION OF THE THEORY. 211

or

l rn

-TTV = /(*) da. (e^ sin x sin a 4- e~2v sin 2x sin 2a
* Jg

+ e~5v sin 3^ sin 3x + &c.)
a is a new variable, which disappears after integration.

If the sum of the series be determined, and if it be substituted
in the last equation, we have the value of v in a finite form. The
double of the series is equal to

e~v [cos (x - a) - cos (x + a)] + e~Zy [cos 2 (x - a) - cos 2 (x + a)]

+ e~zv [cos 3 (a? - a) - cos 3 (x 4 a)] + &c. ;
denoting by F (y,p) the sum of the infinite series

e~v cosp -f e~^ cos 2^ -f e*v cos 3/> -f &a,

we find

TTl1

- f/W ^

-' 0

We have also

,-(v+p\/-i) g-to-pV-i)

J g-(i/+PV-l)

or

F(ytp) = ™*P-«

ev-2cos/?-fe-<"

cos (# 4- a) - e"

cos^?
whence

— 2 cos (a? — ct) 4- e v e1' — 2 cos ^ -L ^ -j- "~v
or

-«) + e^] [ev- 2 cos («
or, decomposing the coefficient into two fractions,

TTU =

fit f -J ^ -i

J o/(«) ^ ^_ 2 cos (a? -*) + £>-" ~ ^-2cos(^+a) + ^j

14—2

212 THEOKY OF HEAT. [CH. III. SECT. VII.

This equation contains, in real terms under a finite form, the
integral of the equation -^ + -=-$ — 0, applied to the problem of

the uniform movement of heat in a rectangular solid, exposed at
its extremity to the constant action of a single source of heat.

It is easy to ascertain the relations of this integral to the
general integral, which has two arbitrary functions; these func
tions are by the very nature of the problem determinate, and
nothing arbitrary remains but the function /(a), considered
between the limits a = 0 and a = ?r. Equation (a) represents,
under a simple form, suitable for numerical applications, the same
value of v reduced to a convergent series.

If we wished to determine the quantity of heat which the solid
contains when it has arrived at its permanent state, we should
take the integral fdxfdy v from x — 0 to x = TT, and from y — 0 to
y = oo ; the result would be proportianal to the quantity required.
In general there is no property of the uniform movement of heat
in a rectangular plate, which is not exactly represented by this
solution.

We shall next regard problems of this kind from another point
of view, and determine the varied movement of heat in different
bodies.

CHAPTER IV.

OF THE LINEAR AND VARIED MOVEMENT OF HEAT IN A RING.

SECTION I.

General solution of the problem.

238. THE equation which expresses the movement of heat
in a ring has been stated in Article 105 ; it is

dv _ K d2v hi ,7 N

dt~Cl)dx*~~CDSV

The problem is now to integrate this equation : we may
write it simply

dv d*v ,

wherein k represents „-= , and h represents yrrTa > x denotes the

length of the arc included between a point m of the ring and the
origin 0, and v is the temperature which would be observed at
the point m after a given time t. We first assume v = e~htufx

7 72 V '

u being a new unknown, whence we deduce -ji = k T~2 » now this

equation belongs to the case in which the radiation is nul at
the surface, since it may be derived from the preceding equa
tion by making h = 0 : we conclude from it that the different
points of the ring are cooled successively, by the action of the
medium, without this circumstance disturbing in any manner the
law of the distribution of the heat.

In fact on integrating the equation -77 = &-TT > we should

dt (tx

find the values of u which correspond to different points of the

214 THEORY OF HEAT. [CHAP. IV.

ring at the same instant, and we should ascertain what the state
of the solid would be if heat were propagated in it without any
loss at the surface ; to determine then what would be the state
of the solid at the same instant if this loss had occurred, it will
be sufficient to multiply all the values of u taken at different
points, at the same instant, by the same fraction e~ht. Thus the
cooling which is effected at the surface does not change the law
of the distribution of heat ; the only result is that the tempera
ture of each point is less than it would have been without this
circumstance, and the temperature diminishes from this cause
according to the successive powers of the fraction e~ht.

239. The problem being reduced to the integration of the

7 72

equation -j- = k ,— 2 , we shall, in the first place, select the sim-
dt dx*

plest particular values which can be attributed to the variable
u ; from them we shall then compose a general value, and we
shall prove that this value is as extensive as the integral, which
contains an arbitrary function of or, or rather that it is this
integral itself, arranged under the form which the problem re
quires, so that there cannot be any different solution.

It may be remarked first, that the equation is satisfied if we
give to u the particular value aemt sin nx, m and n being subject
to the condition m — — Jen*. Take then as a particular value of
u the function e~knH sin nx.

In order that this value may belong to the problem, it must
not change when the distance x is increased by the quantity 2?rr,
r denoting the mean radius of the ring. Hence Zirnr must be a

ft

multiple i of the circumference 2?r ; which gives n = - .

We may take i to be any integer; we suppose it to be
always positive, since, if it were negative, it would suffice to
change the sign of the coefficient a in the value ae~knH sin nx.

_kn fa
The particular value ae r* sin — could not satisfy the problem

proposed unless it represented the initial state of the solid. Now

7 or
on making t = 0, we find u = a sin — : suppose then that the

SECT. I.] PARTICULAR SOLUTIONS. 215

X

initial values of u are actually expressed by a sin-; that is to \

say, that the primitive temperatures at the different points are
proportional to the sines of angles included between the radii Vv
which pass through those points and that which passes through
the origin, the movement of heat in the interior of the ring will

Jet

X

be exactly represented by the equation u — ae r* sin - , and if
we take account of the loss of heat through the surface, we find

-(h + tyt . X

v = ae v *-' sm - .
r

In the case in question, which is the simplest of all those which
we can imagine, the variable temperatures preserve their primi
tive ratios, and the temperature at any point diminishes accord
ing to the successive powers of a fraction which is the same for
every point.

The same properties would be noticed if we supposed the
initial temperatures to be proportional to the sines of the double

/Vl

of the arc - ; and in general the same happens when the given

n -v»

temperatures are represented by a sin — , i being any integer
whatever.

We should arrive at the same results on taking for the
particular value of u the quantity ae~kn2tcos nx : here also we have

•

2mrr = 2«V, and n — - ; hence the equation

-k% ix

u — ae r cos —
r

expresses the movement of heat in the interior of the ring if the

? ¥

initial temperatures are represented by cos — .

In all these cases, where the given temperatures are propor
tional to the sines or to the cosines of a multiple of the arc - ,

the ratios established between these temperatures exist con
tinually during the infinite time of the cooling. The same would

216 THEORY OF HEAT. [CHAP. IV.

be the case if the initial temperatures were represented by the
function a sin 1- b cos — , i being any integer, a and b any co
efficients whatever.

240. Let us pass now to the general case in which the initial
temperatures have not the relations which we have just supposed,
but are represented by any function whatever F(x). Let us give

(x\ I ic\

- ) , so that we have F (as) — <j> ( - j , and

imagine the function </>(-) to be decomposed into a series of

sines or cosines of multiple arcs affected by suitable coefficients.
We write down the equation

*p sin (O - ) + a, sin (l X] + a2 sin (2 *} + &c.
\ r) \ rj \ rj

I

+ &0c

The numbers a0, alt aa ..., 60, ^, 62 ... are regarded as known
and calculated beforehand. It is evident that the value of u will
then be represented by the equation

fc'-

u =*

. X

a, sm -

-L

o, cos -

r* «„ sin 2 -

> cos 2-
2 r

. &c.
x

In fact, 1st, this value of u satisfies the equation -7- = k -7- j,

dt d/x

since it is the sum of several particular values ; 2nd, it does not
change when we increase the distance x by any multiple whatever
of the circumference of the ring ; 3rd, it satisfies the initial state,
since on making t = 0, we find the equation (e). Hence all the
conditions of the problem are fulfilled, and it remains only to
multiply the value of u by e~ht.

241. As the time increases, each of the terms which compose
the value of u becomes smaller and smaller ; the system of tem
peratures tends therefore continually towards the regular and con-

SECT. I.]

COMPLETE SOLUTION.

217

stant state in which the difference of the temperature u from the
constant b0 is represented by

x

x\
- )

a sm - + b cos - e
r rj

Thus the particular values which we have previously considered,
and from which we have composed the general value, derive their
origin from the problem itself. Each of them represents an
elementary state which could exist of itself as soon as it is sup
posed to be formed ; these values have a natural and necessary
relation with the physical properties of heat.

To determine the coefficients «0, alt a2, &c., 60, 61? &2, &c., we
must employ equation (II), Art. 234, which was proved in the
last section of the previous Chapter.

Let the whole abscissa denoted by X in this equation be 2?rr,
let x be the variable abscissa, and let f(x] represent the initial
state of the ring, the integrals must be taken from x = 0 to
x = 2-Trr ; we have then

*) ~ 3 //(

*•

+ sin

in(3/si

sn

Knowing in this manner the values of a0, «1, a2, &c.,
b0, bt, b2, &c., if they be substituted in the equation we have
the following equation, which contains the complete solution of
the problem :

irrv

. x

sm -
r

COS-

kt

x r / 2t£

sin 2 - I ( sin — / (x) a
rj\ r *

cos 2 - ( fcoB— /(a?) dx J

+ &al

(E).

218 THEORY OF HEAT. [CHAP. IV.

All the integrals must be taken from x = 0 to x = 2?rr.

The first term ^ — \f(x] dx> which serves to form the value of

v, is evidently the mean initial temperature, that is to say, that
.which each point would have it" all the initial heat were distri
buted equally throughout.

242. The preceding equation (E) may be applied, whatever
'the form of the given function f(x) may be. We shall consider
two particular cases, namely : 1st, that which occurs when the
ring having been raised by the action of a source of heat to its
permanent temperatures, the source is suddenly suppressed ; 2nd,
the case in which half the ring, having been equally heated
throughout, is suddenly joined to the other half, throughout which
the initial temperature is 0.

k1

We have seen previously that the permanent temperatures

of the ring are expressed by the equation v = azx + bz~x ; the

value of quantity a being e KS , where I is the perimeter of the
generating section, and S the area of that section.

If it be supposed that there is but a single source of heat, the
equation -7- = 0 must necessarily hold at the point opposite to

that which is occupied by the source. The condition aoLx—boTx= 0
will therefore be satisfied at this point. For convenience of calcu
lation let us consider the fraction -yj— to be equal to unity, and let

us take the radius r of the ring to be the radius of the trigono
metrical tables, we shall then have v = aex + be~x ; hence th<£~mitial
state of the ring is represented by the equation

v = le*(e*+*+e™).

It remains only to apply the general equation (E), and de
noting by M the mean initial heat (Art. 241), we shall have

This equation expresses the variable state of a solid ring, which
having been heated at one of its points and raised to stationary

SECT. I.] FURTHER APPLICATION. 219

temperatures/ cools in air after the suppression of the source of
heat.

243. In order to make a second application of the general
equation (E), we shall suppose the initial heat to be so distributed
that half the ring included between x = 0 and x = TT has through
out the temperature 1, the other half having the temperature 0.
It is required to determine the state of the ring after the lapse of
a time t.

The function /(#), which represents the initial state, is in this
case such that its value is 1 so long as the variable is included
between 0 and TT. It follows from this that we must suppose
f(x) = 1, and take the integrals only from x = 0 to x = TT, the
other parts of the integrals being nothing by hypothesis. We
"obtain first the following equation, which gives the development
of the function proposed, whose value is 1 from x = Q to X = TT and
nothing from x = TT to x = 2w,

f(x) = o + — ( sm x + o sin %x + ^ sin oaj + = sin 7-z + &c. ) .

A 7T \ O O / /

If now we substitute in the general equation the values which
we have just found for the constant coefficients, we shall have the
equation

x TTV = e~htt-77r + sin xe~kt + ^$m 3xe~®kt +^ sin oxe~5ZJct + &c
2i \4 o o

which expresses the law according to which the temperature at
each point of the ring varies, and indicates its state after any
given time : we shall limit ourselves to the two preceding applica
tions, and add only some observations on the general solution
expressed by the equation^ (E).

244. 1st. If k is supposed infinite, the state of the ring is
expressed thus, 7rrv = e~ht^lf(x)dx) or, denoting by M the

mean initial temperature (Art. 241), v = e~MM. The temperature
at every point becomes suddenly equal to the mean temperature,
and all the different points retain always equal temperatures,
which is a necessary consequence of the hypothesis in which we
admit infinite conducibility.

220 THEORY OF HEAT. [CHAP. IV.

2nd. We should have the same result if the radius of the ring
were infinitely small.

3rd. To find the mean temperature of the ring after a time t
we must take the integral \f(x)dx from x = 0 to x=%7rr, and

divide by 2?rr. Integrating between these limits the different
parts of the value of u, and then supposing x — 2?rr, we find the
total values of the integrals to be nothing except for the first
term ; the value of the mean temperature is therefore, after the
time t, the quantity e~MM. Thus the mean temperature of the
ring decreases in the same manner as if its conducibility were in
finite ; the variations occasioned by the propagation of heat in the
solid have no influence on the value of this temperature.

In the three cases which we have just considered, the tem
perature decreases in proportion to the powers of the fraction e~h,
or, which is the same thing, to the ordinate of a logarithmic
curve, the abscissa being equal to the time which has elapsed.
This law has been known for a long time, but it must be remarked
that it does not generally hold unless the bodies are of small
dimensions. The previous analysis tells us that if the diameter of
a ring is not very small, the cooling at a definite point would not
be at first subject to that law ; the same would not be the case
with the mean temperature, which decreases always in proportion
to the ordinates of a logarithmic curve. For the rest, it must not
be forgotten that the generating section of the ring is supposed to
have dimensions so small that different points of the same section
do not differ sensibly in temperature.

4th. If we wished to ascertain the quantity of heat which
escapes in a given time through the surface of a given portion of

the ring, the integral hi \ dt I vdx must be employed, and must

be taken between limits relative to the time. For example,
if we took 0 and ZTT to be the limits of x, and 0, oo , to be the
limits of t\ that is to say, if we wished to determine the whole
quantity of heat which escapes from the entire surface, during the
complete course of the cooling, we ought to find after the integra
tions a result equal to the whole quantity of the initial heat, or
QjrrM, M being the mean initial temperature.

SECT. I.] DISTRIBUTION OF HEAT IX THE RING. 221

5th. If we wish to ascertain how much heat flows in a given
time, across a definite section of the ring, we must employ the

integral - KS I dt -=- , writing for -y- the value of that function,
J dx cLx

taken at the point in question.

245. Heat tends to be distributed in the ring according to
a law which ought to be noticed. The more the time which
has elapsed increases the smaller do the terms which compose
the value of v in equation (E) become with respect to those
which precede them. There is therefore a certain value of t for
which the movement of heat begins to be represented sensibly
by the equation

/ x x\ _Tct

u = an+ (a. sin - 4- Z> cos - ) e r- .

\ l r rj

The same relation continues to exist during the infinite time
of the cooling. In this state, if we choose two points of the ring
situated at the ends of the same diameter, and represent their
respective distances from the origin by xv and #2, and their cor
responding temperatures at the time t by vl and vz ; we shall have

Vl = Ja0+ (^ sin^-l-^ cos^-H e~^^~ht

f , t - x* , T, X2\ -
v» ~ 1 ao + ai sm + &i cos e

2 ( ° \ l r rj

The sines of the two arcs — and '-f differ only in sign ; the

or TT

same is the case with the quantities cos — and cos — ; hence

r r

thus the half-sum of the temperatures at opposite points gives
a quantity a0e~ht, which would remain the same if we chose two
points situated at the ends of another diameter. The quantity
a0e~ht, as we have seen above, is the value of the mean tempera
ture after the time t. Hence the half-sum of the temperature
at any two opposite points decreases continually with the mean
temperature of the ring, and represents its value without sensible
error, after the cooling has lasted for a certain time. Let us

222 THEORY OF HEAT. [CHAP. IV.

examine more particularly in what the final state consists, which
is expressed by the equation

f / • X -L X\ --} M

v = -\aQ+ f «j sin - + 6, cos -j e <*> e~ht.

If first we seek the point of the ring at which we have the
condition

/7i \

a, sin - + b cos - = 0, or - = — arc tan ( — ) ,
r r r \aj

we see that the temperature at this point is at every instant
the mean temperature of the ring : the same is the case with
the point diametrically opposite ; for the abscissa x of the latter
point will also satisfy the above equation

IT f r)

- = arc tan I L

r \ a^

Let us denote by X the distance at which the first of these
points is situated, and we shall have

X

sin —

*'=-a'— y;

cos —
r

and substituting this value of blt we have

cos —
r

If we now take as origin of abscissae the point which corre
sponds to the abscissa X, and if we denote by u the new abscissa
x — X, we shall have

= e~ht a + sn - e

At .the origin, where the abscissa u is 0, and at the opposite
point, the temperature v is always equal to the mean tempera
ture ; these two points divide the circumference of the ring into
two parts whose state is similar, but of opposite sign ; each point
of one of these parts has a temperature which exceeds the mean
temperature, and the amount of that excess is proportional to
the sine of the distance from the origin. Each point of the

SECT. 1.] PARTIAL CHANGES OF TEMPERATURE. 223

other part has a temperature less than the mean temperature,
and the defect is the same as the excess at the opposite point.
This symmetrical distribution of heat exists throughout the whole
duration of the cooling. At the two ends of the heated half,
two flows of heat are established in direction towards the cooled
half, and their effect is continually to bring each half of the
ring towards the mean temperature.

246. - We may now remark that in the general equation which
gives the value of v, each of the terms is of the form

x x\ - <> —

a, sin i - + b. cos i - } e l'^.
r r)

We can therefore derive, with respect to each term, consequences
analogous to the foregoing. In fact denoting by X the distance
for which the coefficient

a. sin i — \- b. cos i —
r r

X

is nothing, we have the equation 6. = — at tan i — , and this sub
stitution gives, as the value of the coefficient,

a being a constant. It follows from this that taking the point
whose abscissa is X as the origin of co-ordinates, and denoting
by u the new abscissa x — X, we have, as the expression of the
changes of this part of the value of v, the function

ae~ smi-e

If this particular part of the value of v existed alone, so as to
make the coefficients of all the other parts nul, the state of the
ring would be represented by the function

i" —

ae~hte~

** . ,'.u\

r'2 Sin (l - } ,

\ rj

and the temperature at each point would be proportional to the
sine of the multiple i of the distance of this point from the origin.
This state is analogous to that which we have already described :

224 THEORY OF HEAT. [CHAP. IV.

it differs from it in that the number of points which have always
the same temperature equal to the mean temperature of the ring
is not 2 only, but in general equal to 2i. Each of these points or
nodes separates two adjacent portions of the ring which are in
a similar state, but opposite in sign. The circumference is thus
found to be divided into several equal parts whose state is alter
nately positive and negative. The flow of heat is the greatest
possible in the nodes, and is directed towards that portion which
is in the negative state, and it is nothing at the points which are
equidistant from two consecutive nodes. The ratios which exist
then between the temperatures are preserved during the whole of
the cooling, and the temperatures vary together very rapidly in
proportion to the successive powers of the fraction

If we give successively to i the values 0, 1, 2, 3, &c., we shall
ascertain all the regular and elementary states which heat can
assume whilst it is propagated in a solid ring. When one of these
simple modes is once established, it is maintained of itself, and the
ratios which exist between the temperatures do not change; but
whatever the primitive ratios may be, and in whatever manner
the ring may have been heated, the movement of heat can be de
composed into several simple movements, similar to those which
we have just described, and which are accomplished all together
without disturbing each other. In each of these states the tempe
rature is proportional to the sine of a certain multiple of the dis
tance from a fixed point. The sum of all these partial temperatures,
taken for a single point at the same instant, is the actual tempera
ture of that point. Now some of the parts which compose this
sum decrease very much more rapidly than the others. It follows
from this that the elementary states of the ring which correspond
to different values of i, and whose superposition determines the
total movement of heat, disappear in a manner one after the
other. They cease soon to have any sensible influence on the
value of the temperature, and leave only the first among them to
exist, in which i is the least of all. In this manner we form an
exact idea of the law according to which heat is distributed in
a ring, and is dissipated at its surface. The state of the ring be
comes more and more symmetrical; it soon becomes confounded

SECT. II.] TRANSFER BETWEEN SEPARATE MASSES. 225

with that towards which it has a natural tendency, and which con
sists in this, that the temperatures of the different points become
proportional to the sine of the same multiple of the arc which
measures the distance from the origin. The initial distribution
makes no change in these results.

SECTION II.

Of the communication of heat between separate masses.

247. We have now to direct attention to the conformity of
the foregoing analysis with that which must be employed to de
termine the laws of propagation of heat between separate masses ;
we shall thus arrive at a second solution of the problem of the
movement of heat in a ring. Comparison of the two results will
indicate the true foundations of the method which we have fol
lowed, in integrating the equations of the propagation of heat in
continuous bodies. We shall examine, in the first place, an ex
tremely simple case, which is that of the communication of heat
between two equal masses.

Suppose two cubical masses m and n of equal dimensions and
of the same material to be unequally heated; let their respective
temperatures be a and b, and let them be of infinite conducibility.
If we placed these two bodies in contact, the temperature in each
would suddenly become equal to the mean temperature \ (a + 6).
Suppose the two masses to be separated by a very small interval,
that an infinitely thin layer of the first is detached so as to be
joined to the second, and that it returns to the first immediately
after the contact. Continuing thus to be transferred alternately,
and at equal infinitely small intervals, the interchanged layer
causes the heat of the hotter body to pass gradually into that
which is less heated; the problem is to determine what would be,
after a given time, the heat of each body, if they lost at their sur
face no part of the heat which they contained. We do not suppose
the transfer of heat in solid continuous bodies to be effected in a
manner similar to that which we have just described: we wish
only to determine by analysis the result of such an hypothesis.

Each of the two masses possessing infinite conducibility, the
quantity of heat contained in an infinitely thin layer, is sud-
F. H. - 15

226 THEORY OF HEAT. [CHAP. IV.

denly added to that of the body with which it is in contact; and a
common temperature results which is equal to the quotient of the
sum of the quantities of heat divided by the sum of the masses.
Let ft) be the mass of the infinitely small layer which is separated
from the hotter body, whose temperature is a; let a and ft be the
variable temperatures which correspond to the time t, and whose
initial values are a and Z>. When the layer co is separated from the
mass m which becomes m — &>, it has like this mass the tempera
ture a, and as soon as it touches the second body affected with the
temperature /3, it assumes at the same time with that body a

temperature equal to — — . The layer a, retaining the last

temperature, returns to the first body whose mass is m — co and
temperature a. We find then for the temperature after the second
contact

. /w/3 + aftA

a [m — a)) + — — &>

v ' \ m + co } c:m

or

m m 4- G)

The variable temperatures a. and /3 become, after the interval
dt, a. -f (a — ft} — , and ft -f (a — /3) — ; these values are found by

Tfl> fITb

suppressing the higher powers of co. We thus have

the mass which had the initial temperature (3 has received in one
instant a quantity of heat equal to md@ or (a— ft) co, which has
been lost in the same time by the first mass. We see by this
that the quantity of heat which passes in one instant from the
most heated body into that which is less heated, is, all other things
being equal, proportional to the actual difference of temperature
of the two bodies. The time being divided into equal intervals,
the infinitely small quantity co may be replaced by kdt, k being the
number of units of mass whose sum contains co as many times as

the unit of time contains dt, so that we have - = — We thus

co dt

obtain the equations

dz = -(a-j3)~dt and d& = (a - 0) - dt.

SECT. II.] RECIPROCAL CONDUCIBILITY. 227

248. If \ve attributed a greater value to the volume w, which
serves, it may be said, to draw heat from one of the bodies
for the purpose of carrying it to the other, the transfer would
be quicker ; in order to express this condition it would be
necessary to increase in the same ratio the quantity k which
enters into the equations. We might also retain the value
of G) and suppose the layer to accomplish in a given time a
greater number of oscillations, which again would be indicated
by a greater value of k. Hence this coefficient represents in some
respects the velocity of transmission, or the facility with which
heat passes from one of the bodies into the other, that is to say,
their reciprocal conducibility.

249. Adding the two preceding equations, we have

dz + d/3 = 0,

and if we subtract one of the equations from the other, we have
d*-d/3+2 (a-/3) - rft = 0, and, making a -£ = ;/,

7)1

Integrating and determining the constant by the condition that

_1M

the initial value is a - b, we have y = (a — b) e m . The differ
ence y of the temperatures diminishes as the ordinate of a loga

rithmic curve, or as the successive powers of the fraction e~m •
As the values of a. and /?, we have

1 1 _?£? 1 1 -***

a =-(a + l) ---(a-b} e ™ , ft = - (a + b) + ^ («• - b} e m .

250. In the preceding case, we suppose the infinitely small
mass &>, by means of which the transfer is effected, to be always
the same part of the unit of mass, or, which is the same thing,
we suppose the coefficient k which measures the reciprocal con
ducibility to be a constant quantity. To render the investigation
in question more general, the constant k must be considered
as a function of the two actual temperatures a. and ft. We should

then have the two equations dx. = - (a - ft) — dt, and

15—2

228 THEORY OF HEAT. [CHAP. IV.

<?£=(«-£)-<#,

m

in which k would be equal to a function of a and /?, which we
denote by <f> (a, /?). It is easy to ascertain the law which
the variable temperatures a and /3 follow, when they approach
extremely near to their final state. Let y be a new unknown
equal to the difference between a and the final value which is

^ (a + 6) or c. Let z be a second unknown equal to the difference
2

c — p. We substitute in place of a and /3 their values c — y and
c — 2 ; and, as the problem is to find the values of y and z,
when we suppose them very small, we need retain in the results
of the substitutions only the first power of y and z. We therefore
find the two equations,

k

-dy = -(z-y}^(c-y) c-z)dt

k

and — dz — — (z — y] $(c — y, c — z) dt,
tail

developing the quantities which are under the sign (/> and omit
ting the higher powers of y and z. We find dy=(z — y) — $>dt,

and dz = — (z — y] — <f>dt. The quantity $ being constant, it

7?2>

follows that the preceding equations give for the value of the
difference z — y,& result similar to that which we found above for
the value of a — /3.

From this we conclude that if the coefficient k, which was
at first supposed constant, were represented by any function
whatever of the variable temperatures, the final changes which
these temperatures would experience, during an infinite time,
would still be subject to the same law as if the reciprocal con-
ducibility were constant. The problem is actually to determine
the laws of the propagation of heat in an indefinite number of
equal masses whose actual temperatures are different.

251. Prismatic masses n in number, each of which is equal
to m, are supposed to be arranged in the same straight line,
and affected with different temperatures a, b, c, d, &c. ; infinitely

SECT. II.] EQUAL PRISMATIC MASSES IN LINE. 229

thin layers, each of which has a mass co, are supposed to be
separated from the different bodies except the last, and are.
conveyed in the same time from the first to the second, from
the second to the third, from the third to the fourth, and so
on ; immediately after contact, these layers return to the masses
from which they were separated ; the double movement taking ,
place as many times as there are infinitely small instants dt\ it I
is required to find the law to which the changes of temperature r-
are subject.

Let a, {$,%$,... co, be the variable values which correspond to
the same time t, and which have succeeded to the initial values
a, b, c, d, &c. When the layers co have been separated from the
n — 1 first masses, and put in contact with the neighbouring
masses, it is easy to see that the temperatures become

a(m — co) ft (m — co) -f aco 7 (m — co) + {3co
m — o)' m m

S (m — co) + 70) ma)

m m + co

or,

a, /3 + (a-/3)^, 7+ (£-7)^, * + (7- 8)^, ...

When the layers co have returned to their former places,
we find new temperatures according to the same rule, which
consists in dividing the sum of the quantities of heat by the sum
of the masses, and we have as the values of a, ft, 7, S, &c., after
the instant dt,

7+ £ - 7- 7) > ••• "> + (f - «>) •

The coefficient of — is the difference of two consecutive dif-
m

ferences taken in the succession a, /5, 7, ... -^, co. As to the first
and last coefficients of — , they may be considered also as dif

ferences of the second order. It is sufficient to suppose the term
a to be preceded by a term equal to a, and the term co to be

230 THEORY OF HEAT. [CHAP. IV.

followed by a term equal to ay. We have then, as formerly, on
substituting kdt for &>, the following equations :

252. To integrate these equations, we assume, according to
the known method,

Ajjflj, «2, «3, ... «„, being constant quantities which must be deter
mined. The substitutions being made, we have the following
equations :

k

«ift = -(«»-«i)>

«JA = -{(«s-«a)-(a8-a1)},

k

-

If we regard at as a known quantity, we find the expression
for a2 in terms of av and A, then that of az in a2 and h ; the same
is the case with all the other unknowns, a4, a5, &c. The first and
last equations may be written under the form

m

and «^ = (K+1 - <O - K - Ol-

SECT. II.] FORM OF THE SOLUTION. 231

Retaining the two conditions a = ax and an = a^, the value
of «2 contains the first power of h, the value of a3 contains the
second power of h, and so on up to aB+1, which contains the
nth power of li. This arranged, aa+l becoming equal to an, we
have, to determine h, an equation of the nih degree, and at re
mains undetermined.

It follows from this that we shall find n values for A, and in
accordance with the nature of linear equations, the general value
of a is composed of n terms, so that the quantities a, /5, 7, ... &c.
are determined by means of equations such as

a = a/' + a/e*'' + a, V"' + &c.,
£ = a/* + <e*'< + a8V + &c,
7 = a/< + ay + a8V"' -f &c.

to = a' + «"'< + a V* + &c.

The values of h, ti, A", &c. are n in number, and are equal to
the n roots of the algebraical equation of the nih degree in h,
which has, as we shall see further on, all its roots real.

The coefficients of the first equation alf a/, a", a"', &c., are
arbitrary ; as for th"e coefficients of the lower lines, they are deter
mined by a number n of systems of equations similar to the pre
ceding equations. The problem is now to form these equations.

253. Writing the letter q instead of -j- , we have the fol-

A/

lowing equations

We see that these quantities belong to a recurrent series
whose scale of relation consists of two terms (q + 2) and - 1. We

232 THEORY OF HEAT. [CHAP. IV.

can therefore express the general term am by the equation
am = A sin mu + B sin (m — 1) u,

determining suitably the quantities A, B, and u. First we find
A and B by supposing m equal to 0 and then equal to 1, which
gives a0 = B sin w, and al = A sin it, and consequently

ai • / i\
am = «, sin ?WM r— — sin (m — 1) u.

sin M
Substituting then the values of

a,n> «»-!» <W &C-

in the general equation

«M=«m-lfe + 2)-<V2>

we find

sin mu = (<£ f 2) sin (m — 1) M — sin (m — 2) w,

comparing which equation with the next,

sin mu — 2 cos u sin (m — 1) u — sin (w — 2) u,

which expresses a known property of the sines of arcs increasing
in arithmetic, progression, we conclude that q -f 2 = cos u, or
q = — 2 versin w ; it remains only to determine the value of the
arcw.

The general value of am being

-r-1- [sin ?m£ — sin (m - 1) w],
sin u L

we must have, in order to satisfy the condition an+l=^ an9 the
equation

sin (n -f 1) u — sin u = sin ?m - sin (n — 1) ut

TT

whence we deduce sin nu = 0, or u = i — , TT being the semi-
circumference and i any integer, such as 0, 1, 2, 3, 4, ... (•» — 1) ;
thence we deduce the n values of q or -y- . Thus all the roots

K

of the equation in h, which give the values of h} ti, h", li'\ &c.
are real and negative, and are furnished by the equations

SECT. II.] PARTICULAR TEMPERATURE-VALUES. 233

A==-2-versinfo-V

m \ nj

T o ^ • /-i ^
/*, = — 2 — versin 1 - ,
7>i \ n)

' H C\ "* ' I Ct " \

i = — 2 — versin 2 - ,

Z-

i— 1 v s-\ *v • I/ tv"!

1 1} = - 2 — versin J (n - 1) - }• .

771

Suppose then that we have divided the semi-circumference TT
into n equal parts, and that in order to form u, we take i of those
parts, i being less than n, we shall satisfy the differential equations
by taking al to be any quantity whatever, and making

sin u — sin Qu - ? versin M.

« = «.- = e m ,

sin u

p Sin 2 It — Sin Iw -^versinu

1 sin u

sin 3i£ — sin 2u ~ versin ««

7 = a, : - e

sin w

sin ?m — sin (n — V}u -^ versin w

w = a. : — ^ J— e m

sin u

As there are n different arcs which we may take for u,
namely,

A7T -7T 7T , TN"^

0- , 1 - , 2 - , , (n — i) - ,

71 71 W X 71

there are also n systems of particular values for a, fS, 7, &c.,
and the general values of these variables are the sums of the
particular values.

254 We see first that if the arc u is nothing, the quantities
which multiply a, in the values of a, j3, 7, &c., become all equal

., . sin u — sin Oz« , .. . .

to unity, since : takes the value 1 when the arc u

sin u

vanishes; and the same is the case with the quantities which are

234 THEORY OF HEAT. [CHAP. IV.

found in the following equations. From this. we conclude that
constant terms must enter into the general values of a, A 7, ... &>.

Further, adding all the particular values corresponding to
a, /3, 7, ... &c., we have

sin nu - — verem u

a + /3+7 + &c. = flL — e r ;

1 smu

an equation whose second member is reduced to 0 provided the
arc u does not vanish ; but in that case we should find n to be

the value of — — - . We have then in general
sin u

a + /3 + 7 + &c. = nal ;

now the initial values of the variables being a, b, c, &c., we must
necessarily have

nal = a + b + c + &c. ;

it follows that the constant term which must enter into each of
the general values of

a, ft, 7, ... ft) is - (a + b + c + &c.),

that is to say, the mean of all the initial temperatures.

As to the general values of a, A 7, ... G>, they are expressed
by the following equations :

, Sin U — Sin Ou -^ venin u*

1 sin u

sin u" — sin Ow" - — venm «-

+ &c.,

1 sin 2 M— sin M -^vewiuu

_(a + & + c + &c.) + a1--s— -- «

Sill 2 M' — Sin id - versln u'

Sin 2Z*" — Sin u" -^ vemin u"

CI - r— ^ - e r
sin u

&c,

SECT. II.] GENEKAL TEMPERATURE-VALUES. 235

1 sin 3it — sin 2u -^versm.*

n v sin u

sin 3w' — sin 2w' -'^^

+ c,

sm
sin 3*£"

sin
&c.,

/'sin ??

* / "1 \ \ "^

1 ^ "*

7 /sin «
i /i /

1 O
M' — Sm (n — 1) l^'\ -^versinw'

-t-Oit

I c f Sln *

It" — Sin (?l — 1) U"\ - ™ versin u"

H
+ &c.

sin w" y

255. To determine the constants a, b, c, <#...&c., we must
consider the initial state of the system. In fact, when the time
is nothing, the values of a, /3, 7, &c. must be equal to a, 6, c, &c.;
we have then n similar equations to determine the n constants.
The quantities

sinw — sinOw, sin2w— sinw, sin3w — sin2w, ... , sin nu— sin (n— 1) u,
may be indicated in this manner,

A sin OM, A sin w, A sin 2w, A sin ou, ... A sin (?i — 1) u ;

the equations proper for the determination of the constants are,
if the initial mean temperature be represented by C,

a = (7+ a± + bt + q + &c.

, v A sin u A sin u' A sin u"

u — u -r- CTJ — - —

smw
C_C , a Asin2it

1 sin u'

, A sin 2?(

f C, • : 77 \- OiC.,

sin i^
{ A sin 2?^"

1 r • I • ivr"

ttl sinw
rf , c , fl A sin 3w ,

" O-i - '. /

sin z*
, A sin 3&

sinw" ^
i A sin 3w" o

IV W -f Ctj . -j

sm u
&c.

sin w'

sintc" iC"

236 THEORY OF HEAT. [CHAP. IV.

The quantities a^ bl} q, dlt and C being determined by these
equations, we know completely the values of the variables

a, 0, 7, 3, ...co.

We can in general effect the elimination of the unknowns in
these equations, and determine the values of the quantities
a, b, c, d, &c.; even when the number of equations is infinite ; we
shall employ this process of elimination in the following articles.

256. On examining the equations which give the general
values of the variables a, j3, 7 ...... o>, we see that as the time

increases the successive terms in the value of each variable de
crease very unequally : for the values of u, u, u", u", &c. being

- 7T „ 7T „ 7T , 7T p

1-, 2-, 3-, 4 - , &c.,
n n n n

the exponents versin u, versin u, versin u", versin u", &c.
become greater and greater. If we suppose the time t to be
infinite, the first term of each value alone exists, and the tempera
ture of each of the masses becomes equal to the mean tempera

ture - (a + b + c +...&G.). Since the time t continually increases,

IV

each of the terms of the value of one of the variables diminishes
proportionally to the successive powers of a fraction which, for the

2fc 2Jfc

— — versin u -- versin u'

second term, is e " , for the third term e n , and so on.
The greatest of these fractions being that which corresponds to
the least of the values of u, it follows that to ascertain the law
which the ultimate changes of temperature follow, we need con
sider only the two first terms; all the others becoming incom
parably smaller according as the time t increases. The ultimate
variations of the temperatures a, ft, 7, &c. are therefore expressed
by the following equations :

1 , 1 . Sin U - Sin Qu -versinu

a = - (a + b + c + &c. + a

n sin u

0 1 f - - , Sill 2lt — Sin U ~*~ versin

P=-(a + £ + c + &c.) + <LI - : -- e m

n ^ sin u

1,7 - . S -—

7 = - (a + b + c + &c.) + cfj

^ '

n sm u

SECT. II.] CONCLUDING TEMPERATURES. 237

257. If we divide the semi-circumference into n equal parts,
and, having drawn the sines, take the difference between two
consecutive sines, the n differences are proportional to the co-

_ ™ versin u

efficients of e r' , or to the second terms of the values of
a, @, 7,...&). For this reason the later values of «, & y...w are
such that the differences between the final temperatures and the

mean initial temperature - (a + b + c + &c.) are always propor
tional to the differences of consecutive sines. In whatever
manner the masses have first been heated, the distribution of
heat is effected finally according to a constant law. If we
measured the temperatures in the last stage, when they differ
little from the mean temperature, we should observe that the
difference between the temperature of any mass whatever and the
mean temperature decreases continually according to the succes
sive powers of the same fraction ; and comparing amongst them
selves the temperatures of the different masses taken at the same
instant, we should see that the differences between the actual
temperatures and the mean temperature are proportional to the
differences of consecutive sines, the semi-circumference having
been divided into n equal parts.

258. If we suppose the masses which communicate heat to each
other to be infinite in number, we find for the arc u an infinitely
small value ; hence the differences of consecutive sines, taken on
the circle, are proportional to the cosines of the corresponding

, sin mu — sin (m — l)u. ,

arcs; for : is equal to cos mil, when the

sin \JL

arc u is infinitely small. In this case, the quantities whose tem
peratures taken at the same instant differ from the mean tempera
ture to which they all must tend, are proportional to the cosines
which correspond to different points of the circumference divided
into an infinite number of equal parts. If the masses which
transmit heat are situated at equal distances from each other on
the perimeter of the semi-circumference TT, the cosine of the arc at
the end of which any one mass is placed is the measure of the
quantity by which the temperature of that mass differs yet from
the mean temperature. Thus the body placed in the middle of
all the others is that which arrives most quickly at that mean

238 THEORY OF HEAT. [CHAP. IV.

temperature ; those which are situated on one side of the middle,
all have an excessive temperature, which surpasses the mean
temperature the more, according as they are more distant from
the middle ; the bodies which are placed on the other side, all
have a temperature lower than the mean temperature, and they
differ from it as much as those on the opposite side, but in con
trary sense. Lastly, these differences, whether positive or negative,
all decrease at the same time, proportionally to the successive
powers of the same fraction ; so that they do not cease to be repre
sented at the same instant by the values of the cosines of the
same semi-circumference. Such in general, singular cases ex-
cepted, is the law to which the ultimate temperatures are subject.
The initial state of the system does not change these results. We
proceed now to deal with a third problem of the same kind as the
preceding, the solution of which will furnish us with many useful
remarks.

\.

259. Suppose n equal prismatic masses to be placed at equal
distances on the circumference of a circle. All these bodies,
enjoying perfect conducibility, have known actual temperatures,
different for each of them ; they do not permit any part of the
heat which they contain to escape at their surface ; an infinitely
thin layer is separated from the first mass to be united to the
second, which is situated towards the right ; at the same time a
parallel layer is separated from the second mass, carried from left
to right, and joined to the third; the same is the case with all the
other masses, from each of which an infinitely thin layer is sepa
rated at the same instant, and joined to the following mass.
Lastly, the same layers return immediately afterwards, and are
united to the bodies from which they had been detached.

Heat is supposed to be propagated between the masses by
means of these alternate movements, which are accomplished
twice during each instant of equal duration; the problem is to
find according to what law the temperatures vary : that is to say,
the initial values of the temperatures being given, it is required to^
ascertain after any given time the new temperature of each of the
masses.

We shall denote by aiy az, aJz,...ai...oJn the initial temperatures
whose values are arbitrary, and by av a2, as...ai...&n the values of

SECT. II.] EQUAL PRISMATIC MASSES IN CIRCLE. 239

the same temperatures after the time t has elapsed. Each of the
quantities a is evidently a function of the time t and of all the
initial values alf az, a3...an: it is required to determine the
functions a.

260. We shall represent the infinitely small mass of the layer
which is carried from one body to the other by a). We may
remark, in the first place, that when the layers have been separated
from the masses of which they have formed part, and placed re
spectively in contact with the masses situated towards the right,
the quantities of heat contained in the different bodies become
(ra — G>) at + a>an, (m — CD) «2 -f a>zv (in — o>) a3 + coy2, . . ., (m — a>) an
+ w^n-i > dividing each of these quantities of heat by the mass m,
we have for the new values of the temperatures

a* + (**-t ~ Gi)'* • • and a« + (a«-l ~ a«) ;

// V i/V

that is to say, to find the new state of the temperature after the
first contact, we must add to the value which it had formerly the

product of — by the excess of the temperature of the body

from which the layer has been separated over that of the body to
which it has been joined. By the same rule it is found that the
temperatures, after the second contact, are

The time being divided into equal instants, denote by dt the
duration of the instant, and suppose o> to be contained in k
units of mass as many times as dt is contained in the units of
time, we thus have a> = kdt. Calling Ja,, da2, (fa3...da.,...cfaH the

240 THEORY OF HEAT. [CHAP. IV.

infinitely small increments which the temperatures a15 «2,...a4...an
receive during the instant dt, we have the following differential
equations :

Ja2 = -dt

k
d*i = -dt

^-i = -^(a«-2-2Vi + <>>

Illi

<fe.^ ~ <&(<.._, -2*. 4 a,).

261. To solve these equations, we suppose in the first place,
according to the known method,

The quantities 6t, 62, &3, ... &„ are undetermined constants, as
also is the exponent li. It is easy to see that the values of
cij, ff2,...«B satisfy the differential equations if they are subject to
the following conditions :

(6 -26,

7/i

Let £ = -v- , we have, beginning at the last equation,

. = 6.-, (2 + 2) -J»

SECT. II.] PARTICULAR SOLUTION. 241

It follows from this that we may take, instead of b1,bz)b3,...
Z>4.,...6n, the n consecutive sines which are obtained by dividing the
whole circumference 2?r into n equal parts. In fact, denoting the

T7"

arc 2- by u, the quantities

iv

sin Qu, sin lu, sin 2w, sin 8w, ... , sin (71 — 1) u,

whose number is n, belong, as it is said, to a recurring series
whose scale of relation has two terms, 2 cos u and — 1 : so that
we always have the condition

sin iu = 2 cos u sin (i —l)u — sin (i — 2) u.
Take then, instead of blt b2> bB,... bn, the quantities

sin Ow, sin lu, sin 2w, . . . sin .(» — !) u,
and we have

q + 2 = 2 cos u, q = — 2 versin it, or ^ = — 2 versin — .

Iv

We have previously written q instead of -=—, so that the value

n/

2k 27T

of ^ is -- versin — ; substituting in the equations these values
of bt and h we have

_2A* „. 2JT

a = sin Oue m " '" ^

_ verein

«3 = sm zue " "

an= sm w —

262. The last equations furnish only a very particular solu
tion of the problem proposed ; for if we suppose t = 0 we have, as
the initial values of a1? «2, a3, ... «„, the quantities

sin OM, sin Iw, sin 2u, ... sin (n — 1) M,

which in general differ from the given values alt aa, aa)...an:
but the foregoing solution deserves to be noticed because it ex
presses, as we shall see presently, a circumstance which belongs to
all possible cases, and represents the ultimate variations of the

F. H. 16

242 THEORY OF HEAT. [CHAP. IV.

temperatures. We see by this solution that, if the initial tem
peratures «j, a2, a2, ... an, were proportional to the sines

27T -, 2-7T 027T . . -N 2?T

sm 0 — , sin 1 — , sin 2 — , ... sin (n - 1) — ,
n n n n

they would remain continually proportional to the same sines, and
we should have the equations

, 2& . 2<7T

where h = — versin -
m n

For this reason, if the masses which are situated at equal dis
tances on the circumference of a circle had initial temperatures
proportional to the perpendiculars let fall on the diameter
which passes through the first point, the temperatures would
vary with the time, but remain always proportional to those per
pendiculars, and the temperatures would diminish simultaneously
as the terms of a geometrical progression whose ratio is the

-S versin »

fraction e n n .

263. To form the general solution, we may remark in the
first place that we could take, instead of &15 52, b3, ... bn, the n
cosines corresponding to the points of division of the circumference
divided into n equal parts. The quantities cos Ou, cos \u, cos 2w,...

cos (n — 1) u, in which u denotes the arc — , form also a recurring

Yl

series whose scale of relation consists of two terms, 2 cos u and — 1,
for which reason we could satisfy the differential equations by
means of the following equations,

-— versin «

otj = cos Oue 7 ,

KM

versin u

«2 = cos lue ' ,

Zkt
— — versin u

a = cos 2ue m

«n=r cos (n — l)ue

SECT. II.] OTHER SOLUTIONS. 243

Independently of the two preceding solutions we could select
for the values of bt, bz, 63, ... bn, the quantities

sin0.2w, sinl.2i*, sin2.2w, sin3.2w, ..., sin(ft-l)2w;
or else

cos0.2w, cosl.2w, cos2.2w, cos3.2w, ..., cos(?i — l)2w.

In fact, each of these series is recurrent and composed of n
terms ; in the scale of relation are two terms, 2 cos 2u and — 1 ;
and if we continued the series beyond n terms, we should find n
others respectively equal to the n preceding.

In general, if we denote the arcs

2-7T 2?T 0 2-7T , . 2-7T „

0 — , 1 — , 2 — , ..., (w— 1) — , &c.,

n n1 n ' n

by ult MS, ws, ..., WB, we can take for the values of blt 5g, 63, ... bn
the w quantities,

sin Ow4, sin lw,., sin 2M4, sin 3w4, ..., sin (n — 1) M, ;
or else

cos Qut) cos lit., cos 2ttj, cos SM,, ..., cos (?i — 1) w4.

The value of A corresponding to each of these series is given by the
equation

i 2&

/^ = versm w, .

771

We can give n different values to i, from i = 1 to i = n.

Substituting these values of blf b2, b3 ... bn) in the equations
of Art. 261, we have the differential equations of Art. 260 satisfied
by the following results :

-^ versing -^rn

tfj = sin Ott,« * , or ofj = cos

— — versin MJ

j ™ ',

„ -— versinwj -^

«3 = sin 2t*4« , a = cos 2u,e

• / i \ ~^ versin M* / t \ -^ versin w«

= sin (n — 1) w4e , a7i = cos (n — 1) M4e *

16—2

244 THEORY OF HEAT. [CHAP. IV.

264. The equations of Art. 260 could equally be satisfied by
constructing the values of each one of the variables ax, aa, «8, ... an
out of the sum of the several particular values which have been
found for that variable ; and each one of the terms which enter
into the general value of one of the variables may also be mul
tiplied by any constant coefficient. It follows from this that,
denoting by Av BI} A2, B2, A3, Bs, ...*An) Bn) any coefficients
whatever, we may take to express the general value of one of the
variables, a^j for example, the equation

/ r> \ — ^n versin Mi

ofwi+l == (A i sin mul 4 B^ cos muj e

versin 11%

+ (A* sin mu>, 4- B cos mu») e "

-?** versinw,,

+ (An sin mun + Bn cos mun) e 7<

The quantities AltA^A33... An, J5X, J5a, J58, ... Bn, which
enter into this equation, are arbitrary, and the arcs uit u2,us, ... un
are given by the equations :

A 2?r - 2?r 2?r 27T

^ = 0-, ^2 = 1-, ". = 2-, ..., Wn=(^l)-.

The general values of the variables cfj, aa, a8, ... an are then
expressed by the following equations :

. _

at = (Al sin Owt + Bl cos OuJ e 5

sn w + cos

_ versin «3

sn w + cos *

&c.;

_m versin

«2 = (A^ sin lu^ + B^ cos IttJ e

- — versin w2

4 (A2 sin \uz 4- B2 cos Iw2) e 1

—~ — versin %

+ (Aa sin lus 4 B3 cos lnj e '
+ &c.;

SECT. II.] GENERAL SOLUTION. 245

a3 = (At sin 2t*, 4 Bl cos zty 0

- — versin ?<2

4- (^42 sin 2wa 4 1?3 cos 2w2) e '

- ^— versin */3

4 (J.3 sin 2?/3 4 #3 cos 2wa) e f
+ &c. ;

an = (^ sin (n — 1)^4- ^ cos (n — 1) ut] e m

versin w2

+ [Aa sin (n - 1) wa + B3 cos (?i - 1) u9] e '

- ?** versin «3

4 {-4.§ sin (n — 1) wa 4 Ba cos (*i — 1) iij e '
4&c.

265. If we suppose the time nothing, the values av a2, cr3, . . . an
must become the same as the initial values alt a2,a3, ... an. We
derive from this n equations, which serve to determine the coeffi
cients Av BV-A2, B2, Ay B3 It will readily be perceived that

the number of unknowns is always equal to the number of equa
tions. In fact, the number of terms which enter into the value
of one of these variables depends on the number of different
quantities versin ul} versin w2, versing, &c., which we find on
dividing the circumference 2?r into n equal parts. Now the

2-7T 2-7T 2-7T

number of quantities versin 0— , versin 1 — , versin 2 — , &c.,

n n n

is very much less than n, if we count only those that are
different. Denoting the number n by 2^ 4 1 if it is odd,
and by 2i if it is even, i 4 1 always denotes the number
of different versed sines. On the other hand, when in the

, .... . 2?r . n 2-7T . 0 2-7T p

series of quantities versin 0 — , versin 1 — , versm 2 — , &c.,

n n n

9

we come to a versed sine, versin X — , equal to one of the former

versin V — , the two terms of the equations which contain this

versed sine form only one term ; the two different arcs % and
«x-, which, have the same versed sine, have also the same cosine,
and the sines differ only in sign. It is easy to see that the
arcs Ux and ux>, which have the same versed sine, are such that

246 THEORY OF HEAT. [CHAP. TV.

the cosine of any multiple whatever of WA is equal to the cosine
of the same multiple of wA', and that the sine of any multiple
of % differs only in sign from the sine of the same multiple
of UK. It follows from this that when we unite into one the
two corresponding terms of each of the equations, the two un
knowns A^ and A A', which enter into these equations, are replaced
by a single unknown, namely A^ — A^. As to the two unknown
B^ and BX they also are replaced by a single one, namely J5A + BX :
it follows from this that the number of unknowns is equal in all
cases to the number of equations ; for the number of terms is
always i + 1. We must add that the unknown A disappears of
itself from the first terms, since it is multiplied by the sine of
a nul arc. Further, when the number n is even, there is found
at the end of each equation a term in which one of the unknowns
disappears of itself, since it multiplies a nul sine ; thus the
number of unknowns which enter into the equations is equal
to 2 (i + 1) — 2, when the number n is even ; consequently the
number of unknowns is the same in all these cases as the number
of equations.

266. To express the general values of the temperatures
ai> a2> as "• an> tne fc>reg°mg analysis furnishes us with the equa
tions

/ . 2-f A27T\ -»* verBinO2-?

a = [A. sin 0.0— H-^ cos 0.0— }e m
1 \ n n /

f A 1 27r D i 2lT\

+ M9sm0.1 — +_B2cos0.1 — }e

\ n n J

sin 0 . 2 — + B cos 0 . 2 ~ e

' n n

4 &c.,

«,=(A

+(*

sin 1 . 0 —
n

• i i 2?r
sin 1 . 1 —

%
w

+ # cos 1.0—^
?i y

+ 5,0081.1-1

'71 /

3 n )

. -^versinO2?

| e w

-?** versin 1 SJT

g Hi »

_±^ versin 2 ??
1 g m

&c.,

SECT. II.] FORM OF THE GENERAL SOLUTION. 247

— e

sn

2.1^r + 7?2cos2.1^)/^versinl?
n n J

9.TT 9.ir\ J*M verein 2 ^

n

+ f^9 sin 2 . 2 — + £3 cos 2 . 2 — ) e~ - v
V ' n n y

+ &c,

f A n 27T A 27T) -=*< versin 0 **

= j JjSin (n-1) 0 — + Bicos (» — 1)0 — \e m

.2 sin (n — 1) 1 - — H^2 cos (n — 1)1 — \ e m *

&c

To form these equations, we must continue in each equation

the succession of terms which contain versin 0 — , versin 1 — ,

n n

versin 2 — , &c. until we have included every different versed

sine ; and we must oniit all the subsequent terms, commencing
with that in which a versed sine appears equal to one of the
preceding.

The number of these equations is n. If n is an even number
equal to 2t, the number of terms of each- equation is i + 1 ; if n
the number of equations is an odd number represented by 2/+ 1,
the number of terms is still equal to i + I. Lastly, among the
quantities AI} Blt A2, B^ &c., which enter into these equations,
there are some which must be omitted because they disappear of
themselves, being multiplied by nul sines.

267. To determine the quantities AVB^AVBV.A^BV &c.,
which enter into the preceding equations, we must consider the
initial state which is known : suppose t = 0,' and instead of
alt «2, «3, &c., write the given quantities ax, a2, a3, &c., which are
the initial values of the temperatures. We have then to determine
Alf Blt A9, B2, Aa, B3, &c., the following equations:

248 THEORY OF HEAT. [CHAP. IV.

ax =A1 sin 0.0^"+ A9 sin 0.1 — + A sin 0.2— + &c.
n w ?&

+ B. cos 0 . 0 — + jR, cos 0 . 1 — + J5_ cos 0 . 2 — + &c.
?i w n

t . - _ 2?r , - ^ 2?r . ._ 2?r n

«2 = A1 sin 1 . 0 — + A sin 1 . 1 — + A. sin 1 . 2 — + &c.
n n n

+ &ooai .0— + #2 cos 1 . 1 — + K cos 1 . 2 — + &c.

n n n

«8 = Al sin 2 . 0 2- + 4a sin 2 . 1 — + A8 sin 2 . 2 — + &c.

+ A cos 2 . 0 — + B.2 cos 2 . 1 — + K cos 2 . 2 — + &c.
n n n

sin (w- 1)1 — +-

2?r
cos (n — 1)1 K
n

A3 sin (n — 1) 2 h &c.

5, cos (w- 1)2 — + &c.
w

,fm\

/7T

w - 1) 0 -

268. In these equations, whose number is ??, the unknown
quantities are Alt Blt A2, B2, A5, Bs, &c., and it is required to
effect the eliminations and to find the values of these unknowns.
We may remark, first, that the same unknown has a different
multiplier in each equation, and that the succession of multipliers
composes a recurring series. In fact this succession is that of the
sines of arcs increasing in arithmetic progression, or of the cosines
of the same arcs ; it may be represented by

sin Qu, sin lu, sin 2w, sin 3w, ... sin (n — 1) u,
or by cos Qu, cos lu, cos 2w, cos Su, ... cos (n — I) u.

/2?r\
The arc u is equal to i I — j if the unknown in question is A.+l

or B.+1. This arranged, to determine the unknown Ai+l by means
of the preceding equations, we must combine the succession of
equations with the series of multipliers, sin Ow, sin lu, sin 2u,
sin Su, ... sin (n — l)ut and multiply each equation by the cor
responding term of the series. If we take the sum of the equa-

SECT. II.] DETERMINATION OF COEFFICIENTS. 249

tions thus multiplied, we eliminate all the unknowns, except
that which is required to be determined. The same is the case
if we wish to find the value of Bi+l ; we must multiply each
equation by the multiplier of Bi+1 in that equation, and then take
the sum of all the equations. It is requisite to prove that by
operating in this manner we do in fact make all the unknowns
disappear except one only. For this purpose it is sufficient to shew,
firstly, that if we multiply term by term the two following series

sin Qu, sin lu, sin 2u, sin 3u, ... sin (n — 1) u,
sin Qv, sin lv, sin 2t>, sin 3v, ... sin (n — T)v,
the sum of the products

sin Qu sin Oy + sin lu sin lv + sin 2u, sin 2v + &c.

is nothing, except when the arcs u and v are the same, each
of these arcs being otherwise supposed to be a multiple of a part

of the circumference equal to -- ; secondly, that if we multiply

term by term the two series

cos Qu, cos lu, cos 2u, ... cos (n — 1) u,
cos Qv, cos lv, cos 2v, ... cos (n — 1) v,

the sum of the products is nothing, except in the case when
u is equal to v ; thirdly, that if we multiply term by term the two
series

sin Qu, sin lu, sin 2u, sin Su, ... sin (n — 1) u,
cos Qv, cos lv, cos 2y, cos 3v, ... cos (n — 1) v,
the sum of the products is always nothing.

269. Let us denote by q the arc — , by pq the arc u, and by

vq the arc v ; ft and v being positive integers less than n. The
product of two terms corresponding to the two first series will
be represented by

sin jpq sin jvq, or - cos j (//, - v) q - ^ cosj (> + v)q,
the letter j denoting any term whatever of the series 0, 1, 2, 3...

250 THEORY OF HEAT. [CHAP. IV.

(n — 1); now it is easy to prove that if we give to j its n successive
values, from 0 to (n — 1), the sum

2 cos 0 (jj, — v) q 4- £ cos 1 (fL — v) q + ~ cos 2 (p — v) q

+ = cos 3 (fjL — v) q + . . . + ~ cos (n - 1) (p - v) q

A Z

has a nul value, and that the same is the case with the series
^ cos 0 (JM + v) q + £ cos 1 (p + v) q + ^ cos 2 (p + v) q

+ 2 cos 3 (/A + v) ^ + . . . + g cos (n ~ 1) (^ + ") £•
In fact, representing the arc (p — v)q by or, which is consequently

2-7T

a multiple of — , we have the recurring series

cos Oa, cos 1#, cos 2z, . . . cos (w — 1) a,
whose sum is nothing.

To shew this, we represent the sum by s, and the two terms of
the scale of relation being 2 cos a and — 1, we multiply successively
the two members of the equation

s = cos Oa + cos 2a + cos 3a + . . . + cos (n — 1) a

by — 2 cos a and by + 1 ; then on adding the three equations we
find that the intermediate terms cancel after the manner of re
curring series.

If we now remark that not. being a multiple of the whole cir
cumference, the quantities cos (n — 1.) a, cos (n — 2) a, cos (n — 3) a,
&c. are respectively the same as those which have been denoted
by cos (— a), cos (— 2a), cos (— 3a), ... &c. we conclude that

2s — 25 cos a = 0 ;

thus the sum sought must in general be nothing. In the same
way we find that the sum of the terms due to the development of
\ cos j (IJL -f v) q is nothing. The case in which the arc represented
by a is 0 must be excepted ; we then have 1 - cos a = 0; that is
to say, the arcs it and v are the same. In this case the term
J cos,/ (jj, + v) q still gives a development whose sum is nothing ;

SECT. II.] ELIMINATION. 251

but the quantity J cosj (ft— i>) q furnishes equal terms, each of
which has the value ^ ; hence the sum of the products term by
term of the two first series is i n.

In the same manner we can find the value of the sum of the
products term by term of the two second series, or

S (cosjvq cosjvq) ;

in fact, we can substitute for cos jpq cosjvq the quantity
J cosj (fj, - v) q + % cosj (fjb + v) q,

and we then conclude, as in the preceding case, that 2 Jcos j(^+v)q
is nothing, and that 2,-J cosj (/it — v) q is nothing, except in the case
where //, = v. It follows from this that the sum of the products
term by term of the two second series, or 2(cosj/j,qcosjvq), is
always 0 when the arcs u and v are different, and equal to \n
when u = v. It only remains to notice the case in which the arcs
fiq and vq are both nothing, when we have 0 as the value of

S (sinjfjiq sinjvq),

which denotes the sum of the products term by term of the two
first series.

The same is not the case with the sum 2(cosj/^ cosjvq) taken
when /j.q and vq are both nothing ; the sum of the products term
by term of the two second series is evidently equal to n.

As to the sum of the products term by term of the two series
sin Ou, s'mlu, sin 2u, sin 3u, ... sin (n — 1) u,
cos OM, cos lu, cos 2u, cos 3u, . . . cos (n — 1) ut

it is nothing in all cases, as may easily be ascertained by the fore
going analysis.

270. The comparison then of these series furnishes the follow
ing results. If we divide the circumference 2?r into n equal
parts, and take an arc u composed of an integral number p of
these parts, and mark the ends of the arcs u, 2u, 3u, ... (n— l)u, it
follows from the known properties of trigonometrical quantities
that the quantities

sin Qu, sin lu, sin 2u, sin 3w, ... sin (n — l)u,

252 THEORY OF HEAT. [CHAP. IV.

or indeed

cos Ou, cos Iw, cos 2w, cos 3u, ... cos (n — 1) u,

form a recurring periodic series composed of n terms : if we com-

27T
pare one of the two series corresponding to an arc u or p. —

n

with a series corresponding to another arc v or v — , and

multiply term by term the two compared series, the sum of the
products will be nothing when the arcs u and v are different. If
the arcs u and v are equal, the sum of the products is equal to |-/?,
when we combine two series of sines, or when we combine two
series of cosines ; but the sum is nothing if we combine a series of
sines with a series of cosines. If we suppose the arcs u and v to
be nul, it is evident that the sum of the products term by term is
nothing whenever one of the two series is formed of sines, or when
both are so formed, but the sum of the products is n if the com
bined series both consist of cosines. In general, the sum of the
products term by term is equal to 0, or \n or n ; known formulae
would, moreover, lead directly to the same results. They are pro
duced here as evident consequences of elementary theorems in
trigonometry.

271. By means of these remarks it is easy to effect the elimi
nation of the unknowns in the preceding equations. The unknown
Av disappears of itself through having nul coefficients ; to find B^
we must multiply the two members of each equation by the co
efficient of Bt in that equation, and on adding all the equations
thus multiplied, we find

To determine A2 we must multiply the two members of each
equation by the coefficient of A9 in that equation, and denoting

the arc - - by q, we have, after adding the equations together,

W9

al sin 0^ 4- a2 sin Iq + as sin 2q + . . . -f an sin (n — l)q =
Similarly to determine Ba we have
rtj cos 0^ 4- az cos 1 q + aa cos 2</ f . . . + an cos (n - 1) q = ^ n

SECT. II.] VALUE OF THE COEFFICIENTS. 253

In general we could find each unknown by multiplying the
two members of each equation by the coefficient of the unknown
in that equation, and adding the products. Thus we arrive at the
following results :

— -ftf2 sin I-77 -fasm2-^" +&c. = 2aisin(i-l)l —

n n n ^ ' n

2?r , 2?r - 2?r 2?r

-- + GLCOS! — +a3cos2 — + &c.= 2«i cos(z-l)! —

n n n J n

.2— +a3cos2.2 — -f &c. = 2a, f cos (i-l)2 —
?i ?i } n

+«2sinl.3— + a3sin2.3 — +&c.=So»sin(*-l)3 —

« 71 71 ?i

s^^cos 0.3— +«2cosl.3— +CLCOS2.3— + &c. = 2aicos(i-l)3^'
2 7i 71 n J n

&c ............................................. . ..................... (M).

To find the development indicated by the symbol %, \ve must
give to i its n successive values 1, 2, 3, 4, &c., and take the sum,
in which case we have in general

n . ^ . ,. 1N/. .,N 2?r , n „ ,-> , . - s , . . . 2?r

g^=2a»sin(t-l)(;-l) — and ^B =s2a»odB(i-l)(;-l) — .

If we give to the integer^' all the successive values 1, 2, 3, 4,
&c. which it can take, the two formulae give our equations, and if
we develope the term under the sign 2, by giving to i its n values
1, 2, 3, ... n, we have the values of the unknowns Al9J$lt A2,BZ,
A3, B3, &c.j and the equations (ra), Art. 267, are completely solved.

272. "We now substitute the known values of the coefficients
Alt Blt A2, B2, A3,BS, &c., in equations (/A), Art. 266, and obtain
the following values :

254 THEORY OF HEAT. [CHAP. IV.

a=^N + JV e * versin ^ + JVe * versin ^ + &c.

= o + sn ^ + cos & e

+ (3/2 sin £2 + JV2 cos qj 6 <versin^ + &c.

= N0 + (M, sin 2ql + Nt cos 2^) 6 * versin *

+ (Mz sin 2g2 + ^ cos 2g2) 6 < vershl *« + &c.

«. = JV0 + {,¥, sin ( j - 1) ^ + ^ cos (j - 1) grj e '

+ M sin - 1 + ^" cos - 1 e « versin «« + &c.

«n = io + sn 71 - q, + , cos n - Sl e

+ {Mz sin (» - 1) qz + JV; cos 0 - 1) £2} e * versin «« + &c.
In these equations

_» , 27T 027T Q27T

e = e «, ^»y— , 22 = 2 — > ^=3 — »&c.,

2

2^ .

= — ^ a cos (i — 1) <71}

M. \ = - 5 a£ sin

n

?^

2V

2

= — A&t COS (i — 1J Q ,

1/Q = - 2 «i sin

71

2 ?i

2

= — S (!< COS (^ 1) Q'g,

2
1TQ = - 51 cbi sin

7i

71

&C.

&c.

273. The equations which we have just set down contain the
complete solution of the proposed problem ; it is represented by
the general equation

o,= -2a,+ -sin(j-l)~Sa«sin(i-l)-^-

2 .xSTT^ -.N^Tr"! -^

+ - cos ( i — 1) — 2a» cos (i — 1) — € «»
n ' n ' n\

n n n

(e),

SECT. II.] APPLICATION OF THE SOLUTION. 255

in which only known quantities enter, namely, av a2, a-3 ... an,
which are the initial temperatures, k the measure of the con-
ducibility, m the value of the mass, n the number of masses
heated, and t the time elapsed.

From the foregoing analysis it follows, that if several equal
bodies n in number are arranged in a circle, and, having received
any initial temperatures, begin to communicate heat to each other
in the manner we have supposed ; the mass, of each body being
denoted by m, the time by t, and a certain constant coefficient by
k, the variable temperature of each mass, which must be a function
of the quantities t, m, and k, and of all the initial temperatures,
is given by the general equation (e). We first substitute instead
of j the number which indicates the place of the body whose
temperature we wish to ascertain, that is to say, 1 for the first
body, 2 for the second, &c.; then with respect to the letter i which
enters under the sign 2, we give to it the n successive values
1, 2, 3, ... n, and take the sum of all the terms. As to the
number of terms which enter into this equation, there must be
as many of them as there are different versed sines belonging to
the successive arcs

0^,1^,2^3^ &
n n n n

that is to say, whether the number n be equal to (2\ + 1) or 2\,
according as it is odd or even, the number of terms which enter
into the general equation is always \ + 1.

274. To give an example of the application of this formula,
let us suppose that the first mass is the only one which at first
was heated, so that the initial temperatures av a2, a3 . . . an are all
nul, except the first. It is evident that the quantity of heat
contained in the first mass is distributed gradually among all the
others. Hence the law of the communication of heat is expressed
by the equation

1 2

«. = - a t -r - a. cos ( j — 1) — e m

} n * n l J n

2 2?T

+ - at cos (j — 1) 2 — e

2 . 02?T -^

+ - «j cos (j— 1)3 — e m n + &c.

tv 7&

THEORY OF HEAT. [CHAP. IV.

If the second mass alone had been heated and the tempera
tures «,, a3, «4, ... an were nul, we should have

2 + - a2 jsin (j - 1) — sin —

2vr 2?r) — —
•+ cos (/—I) — cos — ^e «"
Vl/ 7 n w I

Bin2

n

^

+ cos (7 -1)2 — cos 2 —
Vi/

4-&C.,

and if all the initial temperatures were supposed nul, except
«t and a2, we should find for the value of aj the sum of the values
found in each of the two preceding hypotheses. In general it is
easy to conclude from the general equation (e), Art. 273, that in
order to find the law according to which the initial quantities of
heat are distributed between the masses, we may consider sepa
rately the cases in which the initial temperatures are mil, one only
excepted. The quantity of heat contained in one of the masses
may be supposed to communicate itself to all the others, regarding
the latter as affected with nul temperatures; and having made
this hypothesis for each particular mass with respect to the initial
heat which it has received, we can ascertain the temperature of
any one of the bodies, after a given time, by adding all the
temperatures which the same body ought to have received on
each of the foregoing hypotheses.

275. If in the general equation (e) which gives the value of
ajt we suppose the time to be infinite, we find a,- = - 2 ai} so that

each of the masses has, acquired the mean temperature ; a result
which is self-evident.

As the value of the time increases, the first term - 2 &i

n

becomes greater and greater relatively to the following terms, or
to their sum. The same is the case with the second with respect
to the terms which follow it; and, when the time has become

SECT. II.] LATER TEMPERATURES. 257

considerable, the value of a,- is represented without sensible error
by the equation,

1 2 f 2?r 2?r

a,- = - 2tai 4- - -tain (j -1) — 2af- sin (i — 1) —
n n { ' n ' n

4 cos ( j - 1) — 2a,- cos (i - 1) —
ft ?? ^

Denoting by a and 6 the coefficients of sin ( / - 1) — and of

n

cos (j — 1) — , and the fraction e~~>* m "*" by G>, we have

7i

1 ( 2-7T 9^

o; = - 2«4 4- to sin (j - 1) — 4 6 cos (j - 1) ~
w ( n n

The quantities a and b are constant, that is to say, independent
of the time and of the letter j which indicates the order of the
mass whose variable temperature is a,-. These quantities are the
same for all the masses. The difference of the variable tempera
ture a.j from, the final temperature - 2af decreases therefore for

IV

each of the masses, in proportion to the successive powers of the
fraction &>. Each of the bodies tends more and more to acquire

the final temperature - 2 ait and the difference between that

final limit and the variable temperature of the same body ends
always by decreasing according to the successive powers of a
fraction. This fraction is the same, whatever be the body whose
changes of temperature are considered ; the coefficient of co* or

(a sin Uj 4 & cos HJ), denoting by KJ the arc ( j - 1) -— , may be put

under the form A sin (uj 4- B), taking A and B so as to have
a = A cos B, and b = A sin B. If we wish to determine the
coefficient of to* with regard to the successive bodies whose
temperature is aj+l) aj+2) aj+3> &c., we must add to HJ the arc

- or 2 — , and so on ; so that we have the equations
n n

% - - 20; = A sin (B 4 %) to* + &c.
n

OLJ '. , - - 2af- = A sin [B 4 Uj 4 1 — J at 4- &c.
n \ n /

F. H. 17

258 THEOEY OF HEAT. [CHAP. IV.

^+2_ 2a. = A sm B + Uj + 2 co* + &c.
„ _ _ 2a. = A sin (j3 + Uj + 3 — ) CD* + &c.

276. We see, by these equations, that the later differences
between the actual temperatures and the final temperatures are
represented by the preceding equations, preserving only the first
term of the second member of each equation. These later differ
ences vary then according to the following law : if we consider
only one body, the variable difference in question, that is to say?
the excess of the actual temperature of the body over the final
and common temperature, diminishes according to the successive
powers of a fraction, as the time increases by equal parts ; and, if
we compare at the same instant the temperatures of all the
bodies, the difference in question varies proportionally to the suc
cessive sines of the circumference divided into equal parts. The
temperature of the same body, taken at different successive equal
instants, is represented by the ordinates of a logarithmic curve,
whose axis is divided into equal parts, and the temperature of
each of these bodies, taken at the same instant for all, is repre
sented by the ordinates of a circle whose circumference is divided
into equal parts. It is easy to see, as we have remarked before,
that if the initial temperatures are such, that the differences of
these temperatures from the mean or final temperature are pro
portional to the successive sines of multiple arcs, these differences
will all diminish at the same time without ceasing to be propor
tional to the same sines. This law, which governs also the initial
temperatures, will not be disturbed by the reciprocal action of the
bodies, and will be maintained until they have all acquired a
common temperature. The difference will diminish for each body
according to the successive powers of the same fraction. Such is
the simplest law to which the communication of heat between a
succession of equal masses can be submitted. When this law has
once been established between the initial temperatures, it is main
tained of itself; and when it does not govern the initial tempera
tures, that is to say, when the differences of these temperatures
from the mean temperature are not proportional to successive
sines of multiple arcs, the law in question tends always to be set

SECT. II.] CONTINUOUS MASSES IN A KING. 259

up, and the system of variable temperatures ends soon by coin
ciding sensibly with that which depends on the ordinates of a
circle and those of a logarithmic curve.

Since the later differences between the excess of the tempera
ture of a body over the mean temperature are proportional to
the sine of the arc at the end of which the body is placed, it
follows that if we regard two bodies situated at the ends of the
same diameter, the temperature of the first will surpass the mean
and constant temperature as much as that constant temperature
surpasses the temperature of the second body. For this reason, if
we take at each instant the sum of the temperatures of two
masses whose situation is opposite, we find a constant sum, and
this sum has the same value for any two masses situated at the
ends of the same diameter.

277. The formulae which represent the variable temperatures
of separate masses are easily applied to the propagation of heat
in continuous bodies. To give a remarkable example, we will
determine the movement of heat in a ring, by means of the
general equation which has been already set down.

Let it be supposed that n the number of masses increases suc
cessively, and that at the same time the length of each mass
decreases in the same ratio, so that the length of the system has
a constant value equal to 2?r. Thus if n the number of masses
be successively 2, 4, 8, 16, to infinity, each of the masses will

be TT, -^, -r, - &c. It must also be assumed that the

£t 4 O

facility with which heat is transmitted increases in the same
ratio as the number of masses in\ thus the quantity which k
represents when there are only two masses becomes double when
there are four, quadruple when there are eight, and so on.
Denoting this quantity by g, we see that the number k must be
successively replaced by g, 2g, 4<g, &c. K we pass now to the
hypothesis of a continuous body, we must write instead of m, the

value of each infinitely small mass, the element dx ; instead of n,

2_
the number of masses, we must write ^— ; instead of k write

n

17—2

260 THEORY OF HEAT. [CHAP. IV.

As to the initial temperatures alt a2, a3...an, they depend on
the value of the arc x, and regarding these temperatures as the
successive states of the same variable, the general value at repre
sents an arbitrary function of x. The index i must then be

x
replaced by -y- . With respect to the quantities alt ag, a3, ...,

these are variable temperatures depending on two quantities
x and t Denoting the variable by v, we have v = $ (x, t). The
index jt which marks the place occupied by one of the bodies,

99

should be replaced by -y-. Thus, to apply the previous analysis to

the case of an infinite number of layers, forming a continuous
body in the form of a ring, we must substitute for the quanti
ties n, m, Ic, ait i, aj} /, their corresponding quantities, namely,

-y- , dx, ff . f(x\ -j- , 4> (x. t\ -7- . Let these substitutions be
dx' ' dx' J ^ J) dx Y ^ •' " dx

made in equation (e) Art. 273, and let ^ dx* be written instead
of versin dx, and i and j instead of i — 1 and j — 1. The first

term - 2o( becomes the value of the integral ~— \f(x) dx taken from
n %Tr)J

07 = 0 to «7=27r; the quantity sm(j-l)^ becomes smjdx or

n

sin x ; the value of cos (/—I) -y- is cos x ; that of - 2a4 sin (i — 1) — -

dx ft n

is -\ f(x] sin JPC&P, the integral being taken from x = 0 to x=2jr :
irj

and the value of - 2a^ cos (i - 1) - r is -If («) cos # cZx, the

integral being taken between the same limits. Thus we obtain
the equation

-f - ( sin x I / (x) sin xdx -f cos x If (x} cos xdx }e-ffnt
fn'\j J /

4- - f sin 2# lf(x)sinZ

cos
(E)

SECT. II.] REMARKS. 261

and representing the quantity gir by k, we have

= g \f(x)dx+ ( sin x \f(x) sin xdx + cos x I /(a;) cos #cta J e w

+ (sin 20ma) sin 2#efo+cos2# //(#) cos 2# dxj e~^kt

TTV

+ &c.

278. This solution is the same as that which was given in the
preceding section, Art. 241 ; it gives rise to several remarks. 1st.
It is not necessary to resort to the analysis of partial differential
equations in order to obtain the general equation which expresses
the movement of heat in a ring. The problem may be solved for f
a definite number of bodies, and that number may then be sup- \
posed infinite. This method has a clearness peculiar to itself, and
guides our first researches. It is eas^afterwards to pass to a
more concise method by a process indicated naturally. We see
that the discrimination of the particular values, which, satisfying
the partial differential equation, compose the general value, is
derived from the known rule for the integration of linear differ
ential equations whose coefficients are constant. The discrimina
tion is moreover founded, as we have seen above, on the physical
conditions of the problem. 2nd. To pass from the case of separate
masses to that of a continuous body, we supposed the coefficient Jc
to be increased in proportion to n, the number of masses. This
continual change of the number k follows from what we have
formerly proved, namely, that the quantity of heat which flows
between two layers of the same prism is proportional to the value

of y- , x denoting the abscissa which corresponds to the section,

and v the temperature. If, indeed, we did not suppose the co
efficient k to increase in proportion to the number of masses, but
were to retain a constant value for that coefficient, we should
find, on making n infinite, a result contrary to that which is
observed in continuous bodies. The diffusion of heat would be
infinitely slow, and in whatever manner the mass was heated, the
temperature at a point would suffer no sensible change during
a finite time, which is contrary to fact. Whenever we resort to
the consideration of an infinite number of separate masses which

262 THEORY OF HEAT. [CHAP. IV.

transmit heat, and wish to pass to the case of continuous bodies,
we must attribute to the coefficient k, which measures the yj^ocity
of transmission, a value proportional to the number of infinitely
small masses which compose the given body.

3rd. If in the last equation which we obtained to express the
value of v or <£ (#, i), we suppose t = 0, the equation necessarily
represents the initial state, we have therefore in this way the
equation (p), which we obtained formerly in Art. 233, namely,

+ sin as I f(x) sin x dx + sin 2# I f(x) sin 2# dx + &c.

(*)<fo J J

+ cos x \ f(x] cos xdx+ cos 2x I f(x) cos 2a? dx + &c.

Thus the theorem which gives, between assigned limits, the
development of an arbitrary function in a series of sines or cosines
of multiple arcs is deduced from elementary rules of analysis.
Here we find the origin of the process which we employed to
make all the coefficients except one disappear by successive in
tegrations from the equation

-f a^ sin x + a^ sin 2# + az sin 3# + &c.
^ * ' ~~ + bt cos x + 5a cos 2x + b3 cos 3# + &c.

These integrations correspond to the elimination of the different
unknowns in equations (m), Arts. 267 and 271, and we see clearly
by the comparison of the two methods, that equation (B), Art. 279,
holds for all values of x included between 0 and 2?r, without its
being established so as to apply to values of x which exceed those
limits.

279. The function 0 (x, t) which satisfies the conditions of
the problem, and whose value is determined by equation (E),
Art. 277, may be expressed as follows :

+ {2sin3ic |^a/(a)sin3a4-2cos3^pa/(a)cos3a}e"32^-f- &c.

SECT. II.] FUNCTIONAL EXPRESSION. 2G3

or 27T$ (x, t} = Idxfty {I + (2 sin x sin a. + 2 cos x cos a) e~w

+ (2 sin 2x sin 2a + 2 cos 2x cos 2a) e~22k*

+ (2 sin 3# sin 3a + 2 cos 3# cos 3 a) e"3^ + &c.}

= fda/(a) [1 + 22 cos i (a - a?) e'**^.

The sign 2 affects the number i, and indicates that the sum
must be taken from 4 = 1 to i = oo . We can also include the
first term under the sign 2, and we have

a?, 0 = cfa/(a) 2 cos / (a - a?) <r'X

We must then give to i all integral values from — co to + oc ;
which is indicated by writing the limits — oo and + oo next to the
sign 2, one of these values of i being 0. This is the most concise
expression of the solution. To develope the second member of the
equation, we- suppose 4 = 0, and then i= 1, 2, 3, &c., and double
each result except the first, which corresponds to i = 0. When
t is nothing, the function <£ (x, t) necessarily represents the initial
state in which the temperatures are equal to / (x), we have there
fore the identical equation,

(B).

We have attached to the signs I and 2 the limits between

which the integral sum must be taken. This theorem holds
generally whatever be the form of the function / (x) in the in
terval from x = 0 to x = 2?r ; the same is the case with that which
is expressed by the equations which give the development of F (x\
Art. 235; and we shall see in the sequel that we can prove directly
the truth of equation (B) independently of the foregoing con
siderations.

280. It is easy to see that the problem admits of no solution
different from that given by equation (E), Art. 277. The function
</> (x, t) in fact completely satisfied the conditions of the problem,

and from the nature of the differential equation -=- = k -, , no

dt da?

264 THEORY OF HEAT. [CHAP. IV.

other function can enjoy the same property. To convince our
selves of this we must consider that when the first state of the

solid is represented by a given equation v1=f(x)t the fluxion -y1
is known, since it is equivalent to k ^ \ . Thus denoting by

#2 or v1-\-Jc -j-1 dt, the temperature at the commencement of the

second instant, we can deduce the value of v2 from the initial
state and from the differential equation. We could ascertain in
the same manner the values va, v4, ... vn of the temperature at
any point whatever of the solid at the beginning of each instant.
Now the function <£ (x, i) satisfies the initial state, since we have
<f) (x, 0) =/(#). Further, it satisfies also the differential equation ;
consequently if it were differentiated, it would give the same

values for -— , -=f , -=/ , &c., as would result from successive
at at at

applications of the differential equation (a). Hence, if in the
function $ (x, t) we give to t successively the values 0, ft), 2o>,
3ft), &c., ft) denoting an element of time, we shall find the same
values vlt vzi vs, &c, as we could have derived from the initial

state by continued application of the equation -y- = k -j—2 . Hence

at doo

every function ^r (x, f) which satisfies the differential equation and
the initial state necessarily coincides with the function <f> (x, t) :
for such functions each give the same function of x, when in them
we suppose t successively equal to 0, co, 2&>, 3&) ... iw, &c.

We see by this that there can be only one solution of the
problem, and that if we discover in any manner a function ^ (x, t)
which satisfies the differential equation and the initial state, we
are certain that it is the same as the former function given by
equation (E).

281. The same remark applies to all investigations whose
object is the varied movement of heat; it follows evidently from
the very form of the general equation.

For the same reason the integral of the equation -rr = k ^
can contain only one arbitrary function of x. In fact, when a

SECT. II.] GENERAL INTEGRAL. 26o

value of v as a function of x is assigned for a certain value of
the time t, it is evident that all the other values of v which
correspond to any time whatever are determinate. We may
therefore select arbitrarily the function of x, which corresponds
to a certain state, and the general function of the two variables
x and t then becomes determined. The same is not the case

with the equation -^ + -7-5 = 0, which was employed in the

preceding chapter, and which belongs to the constant movement
of heat ; its integral contains two arbitrary functions of x and y :
but we may reduce this investigation to that of the varied move
ment, by regarding the final and permanent state as derived from
the states which precede it, and consequently from the initial
state, which is given.

The integral which we have given

~ (dzf (a) 2e - m cos * (a - a?)

contains one arbitrary function f(x), and has the same extent as
the general integral, which also contains only one arbitrary func
tion of x ; or rather, it is this integral itself arranged in a form
suitable to the problem. In fact, the equation v1 =f (x} represent
ing the initial state, and v = <f> (x, t) representing the variable
state which succeeds it, we see from the very form of the heated
solid that the value of v does not change when x ± i%7r is written
instead of x, i being any positive integer. The function

^e-izkt cosl' (a — #)

satisfies this condition; it represents also the initial state when
we suppose t = 0, since we then have

(a) X cos i (a — x),

an equation which was proved above, Arts. 235 and 279, and is
also easily verified. Lastly, the same function satisfies the differ
ential equation -=- = k -5-5 . Whatever be the value of t, the

temperature v is given by a very convergent series, and the different
terms represent all the partial movements which combine to form

266 THEORY OF HEAT. [CHAP. IV.

the total movement. As the time increases, the partial states of
higher orders alter rapidly, but their influence becomes inappre
ciable; so that the number of values which ought to be given to
the exponent i diminishes continually. After a certain time the
system of temperatures is represented sensibly by the terms which
are found on giving to i the values 0, + 1 and ± 2, or only 0

and ± 1, or lastly, by the first of those terms, namely, ~— I da/ (at) ;

there is therefore a manifest relation between the form of the
solution and the progress of the physical phenomenon which has
been submitted to analysis.

282. To arrive at the solution we considered first the simple
values of the function v which satisfy the differential equation :
we then formed a value which agrees with the initial state, and
has consequently all the generality which belongs to the problem.
We might follow a different course, and derive the same solution
from another expression of the integral ; when once the solution
is known, the results are easily transformed. If we suppose the
diameter of the mean section of the ring to increase infinitely, the
function <£ (a?, t), as we shall see in the sequel, receives a different
form, and coincides with an integral which contains a single
arbitrary function under the sign of the definite integral. The
latter integral might also be applied to the actual problem; but,
if we were limited to this application, we should have but a very
imperfect knowledge of the phenomenon; for the values of the
temperatures would not be expressed by convergent series, and
we could not discriminate between the states which succeed each
other as the time increases. The periodic form which the problem
supposes must therefore be attributed to the function which re
presents the initial state; but on modifying that integral in this
manner, we should obtain no other result than

0 0> 0 = IT- {<**/ (») 2e-**« cos i (OL - x).

ATTJ

From the last equation we pass easily to the integral in
question, as was proved in the memoir which preceded this work.
It is not less easy to obtain the equation from the integral itself.
These transformations make the agreement of the analytical
results more clearly evident ; but they add nothing to the theory,

SECT. II.] DIFFERENT INTEGRAL FORMS. 2G7

and constitute no different analysis. In oneofthe following
chapters we shall examine the different forms whicfT may be

assumed by the integral of the equation -r — ^-r^^ the relations

dv dx

which they have to each other, and the cases in which they ought
to be employed.

To form the integral which expresses the movement of heat in
a ring, it was necessary to resolve an arbitrary function into a
series of sines and cosines of multiple arcs; the numbers which
affect the variable under the symbols sine and cosine are the
natural numbers 1, 2, 3, 4, &c. In the following problem the
arbitrary function is again reduced to a series of sines; but the
coefficients of the variable under the symbol sine are no longer
the numbers 1, 2, 3, 4, &c.: these coefficients satisfy a definite
equation whose roots are all incommensurable and infinite in
number.

Note on Sect. I, Chap. IV. Guglielmo Libri of Florence was the first to
investigate the problem of the movement of heat in a ring on the hypothesis of
the law of cooling established by Dulong and Petit. See his Memoire sur la
theorie de la chaleur, Crelle's Journal, Band VII., pp. 116—131, Berlin, 1831.
(Read before the French Academy of Sciences, 1825. ) M. Libri made the solution
depend upon a series of partial differential equations, treating them as if they
were linear. The equations have been discussed in a different manner by
Mr Kelland, in his Theory of Heat, pp. 69 — 75, Cambridge, 1837. The principal
result obtained is that the mean of the temperatures at opposite ends of any
diameter of the ring is the same at the same instant. [A. F.]

CHAPTER V.

OF THE PROPAGATION OF HEAT IN A SOLID SPHERE.
SECTION I.

General solution.

283. THE problem of the propagation of heat in a sphere has
been explained in Chapter II., Section 2, Article 117; it consists
in integrating the equation

dv , fd*v 2 dv\
so that when x — X the integral may satisfy the condition

,
ax

k denoting the ratio ™ , and h the ratio -^ of the two con-

ducibilities ; v is the temperature which is observed after
the time t has elapsed in a spherical layer whose radius is a?;
X is the radius of the sphere ; v is a function of x and t, which is
equal to F (x) when we suppose * = 0. The function F(x) is
given, and represents the initial and arbitrary state of the solid.
If we make y = vx, y being a new unknown, we have,

after the substitutions, ^f = ^T^: tnus we must integrate the

last equation, and then take »«•', We shall examine, in the

sc

first place, what are the simplest values which can be attributed
to if, and then form a general value which will satisfy at the same

CHAP. V. SECT. I.] PARTICULAR SOLUTIONS. 269

time the differential equation, the condition relative to the
surface, and the initial state. It is easily seen that when these
three conditions are fulfilled, the solution is complete, and no
other can be found.

284. Let y — emtu, u being a function of x, we have

First, we notice that when the value of t becomes infinite, the
value of v must be nothing at all points, since the body is com
pletely cooled. Negative values only can therefore be taken for
m. Now k has a positive numerical value, hence we conclude
that the value of u is a circular function, which follows from the
known nature of the equation

, <Fu

mu = k -j-s .
dx

Let u = A cos nx + B sin nx • we have the condition m = — k w2.
Thus we can express a particular value of v by the equation

e-knH

v = - (A cos nx -f B sin nx\
so

where n is any positive number, and A and B are constants. We
may remark, first, that the constant A ought to be nothing ; for
the value of v which expresses the temperature at the centre,
when we make x = 0, cannot be infinite ; hence the term A cos nx
should be omitted.

Further, the number n cannot be taken arbitrarily. In fact,
if in the definite equation -j- + hv — 0 we substitute the value

of v, we find

nx cos nx + (hoc — 1) sin nx = 0.

As the equation ought to hold at the surface, we shall suppose
in it x = X the radius of the sphere, which gives

Let X be the number 1 — hX> and nX — e, we have - - = X.

tan e

We must therefore find an arc 6, which divided by its tangent

270

THEORY OF HEAT.

[CHAP. V.

gives a known quotient X, and afterwards take n = -^ . It is

JL

evident that there are an infinity of such arcs, which have a given
ratio to their tangent ; so that the equation of condition

nX - I _ XT

, -vr- — -L m.\.

tan nX
has an infinite number of real roots.

285. Graphical constructions are very suitable for exhibiting
the nature of this equation. Let u = tan e (fig. 12), be the equation

Fig. 12.

to a curve, of which the arc e is the abscissa, and u the ordinate ;
and let u = - be the equation to a straight line, whose co-ordinates

A

are also denoted by e and u. If we eliminate u from these two
equations, we have the proposed equation - = tan e. The un-

A

known e is therefore the abscissa of the point of intersection of
the curve and the straight line. This curved line is composed of
an infinity of arcs ; all the ordinates corresponding to abscissae

1357

271"' 271"' 271"' 271"'

are infinite, and all those which correspond to the points 0, TT,
27T, STT, &c. are nothing. To trace the straight line whose

. 6 €

equation is u — - = j-^f we form the square oi coi, and

A, 1 — ilJL

measuring the quantity hX from co to h, join the point h with
the origin 0. The curve non whose equation is u—tsm e has for

SECT. I.] ROOTS OF EQUATION OF CONDITION. 271

tangent at the origin a line which divides the right angle into two
equal parts, since the ultimate ratio of the arc to the tangent is 1.
We conclude from this that if X or 1—TiX is a quantity less than
unity, the straight line mom passes from the origin above the
curve non, and there is a point of intersection of the straight line
with the first branch. It is equally clear that the same straight
line cuts all the further branches mrn, H^TTH, &c. Hence the

equation = X has an infinite number of real roots. The

tan e

first is included between 0 and ^, the second between TT and

— , the third between STT and -^- , and so on. These roots
2t *2*

approach very near to their upper limits when they are of a very
advanced order.

286. If we wish to calculate the value of one of the roots,
for example, of the first, we may employ the following rule : write

down the two equations e = arc tan u and u = - , arc tan u de*

A<

noting the length of the arc whose tangent is u. Then taking
any number for u, deduce from the first equation the value of e ;
substitute this value in the second equation, and deduce another
value of u ; substitute the second value of u in the first equation ;
thence we deduce a value of 6, which, by means of the second
equation, gives a third value of u. Substituting it in the first
equation we have a new value of e. Continue thus to determine
u by the second equation, and e by the first. The operation gives
values more and more nearly approaching to the unknown e, as is
evident from the following construction.

In fact, if the point u correspond (see fig. 13) to the arbitrary
value which is assigned to the ordinate u ; and if we substitute
this value in the first equation e = arc tan u, the point e will
correspond to the abscissa which we have calculated by means
of this equation. If this abscissa e be substituted in the second

equation u = - , we shall find an ordinate u which corresponds

to the point u. Substituting u in the first equation, we find an
abscissa e' which corresponds to the point e' ; this abscissa being

272

THEORY OF HEAT.

[CHAP. V.

then substituted in the second equation gives rise to an ordinate
w', which when substituted in the first, gives rise to a third
abscissa e", and so on to infinity. That is to say, in order to
represent the continued alternate employment of the two pre-

Fig. 13.

Fig. 14.

ceding equations, we must draw through the point u a horizontal
line up to the curve, and through e the point of intersection draw
a vertical as far as the straight line, through the point of inter
section u draw a horizontal up to the curve, through the point of
intersection e' draw a vertical as far as the straight line, and so on
to infinity, descending more and more towards the point sought.

287. The foregoing figure (13) represents the case in which
the ordinate arbitrarily chosen for u is greater than that which
corresponds to the point of intersection. If, on the other hand, we
chose for the initial value of u a smaller quantity, and employed

in the same manner the two equations e = arc tan u, u — - , we

A

should again arrive at values successively closer to the unknown
value. Figure 14 shews that in this case we rise continually
towards the point of intersection by passing through the points
ueu e u" e", &c. which terminate the horizontal and vertical lines.
Starting from a value of u which is too small, we obtain quantities
e e' e" e'", &c. which converge towards the unknown value, and are
smaller than it ; and starting from a value of u which is too great,
we obtain quantities which also converge to the unknown value,
and each of which is greater than it. We therefore ascertain

SECT. I.] MODE OF APPROXIMATION. 273

successively closer limits between the which magnitude sought is
always included. Either approximation is represented by the
formula

= . . . arc tan

- arc tan j - arc tan f- arc tan - ) I \.

When several of the operations indicated have been effected,
the successive results differ less and less, and we have arrived at
an approximate value of e.

288. We might attempt to apply the two equations
e = arc tan u and u = -

A.

in a different order, giving them the form u = tan e and e = \n.
We should then take an arbitrary value of e, and, substituting it
in the first equation, we should find a value of u, which being
substituted in the second equation would give a second value of
e; this new value of e could then be employed in the same
manner as the first. But it is easy to see, by the constructions
of the figures, that in following this course of operations we
depart more and more from the point of intersection instead of
approaching it, as in the former case. The successive values of e
which we should obtain would diminish continually to zero, or
would increase without limit. We should pass successively from
e" to u", from u" to e', from e to u', from u' to e, and so on to
infinity.

The rule which we have just explained being applicable to the
calculation of each of the roots of the equation

tan e

which moreover have given limits, we must regard all these roots
as known numbers. Otherwise, it was only necessary to be as
sured that the equation has an infinite number of real roots.
We have explained this process of approximation because it is
founded on a reinarkable construction, which may be usefully
employed in several cases, and which exhibits immediately the
nature and limits of the roots ; but the actual application of the
process to the equation in question would be tedious ; it would be
easy to resort in practice to some other mode of approximation.

F. H. 18

274 THEOKY OF HEAT. [CHAP. V.

289. We now know a particular form which may be given to
the function v so as to satisfy the two conditions of the problem.
This solution is represented by the equation

Ae~knH sin nx , . sin nx

v — - or v —

, 2.

— -Kn t

x nx

The coefficient a is any number whatever, and the number n is

n X
such that - - Tr=l—hX. It follows from this that if the

initial temperatures of the different layers were proportional to

the quotient - - — , they would all diminish together, retaining
fix

between themselves throughout the whole duration of the cooling
the ratios which had been set up ; and the temperature at each
point would decrease as the ordinate of a logarithmic curve whose
abscissa would denote the time passed. Suppose, then, the arc e
being divided into equal parts and taken as abscissa, we raise at
each point of division an ordinate equal to the ratio of the sine to
the arc. The system of ordinates will indicate the initial tem
peratures, which must be assigned to the different layers, from the
centre to the surface, the whole radius X being divided into equal
parts. The arc e which, on this construction, represents the
radius X, cannot be taken arbitrarily; it is necessary that the
arc and its tangent should be in a given ratio. As there are
an infinite number of arcs which satisfy this condition, we might
thus form an infinite number of systems of initial temperatures,
which could exist of themselves in the sphere, without the ratios
of the temperatures changing during the cooling.

290. It remains only to form any initial state by means of
a certain number, or of an infinite number of partial states, each
of which represents one of the systems of temperatures which we
have recently considered, in which the ordinate varies with the
distance x, and is proportional to the quotient of the sine by the
arc. The general movement of heat in the interior of a sphere
will then be decomposed into so many particular movements, each
of which is accomplished freely, as if it alone existed.

Denoting by nlt na, n3, &c., the quantities which satisfy the

equation - - ^=1 — hX, and supposing them to be arranged in

SECT. I.] COEFFICIENTS OF THE SOLUTION. 275

order, beginning with the least, we form the general equa
tion

vx = a£~ltn?i sin njc + a2e~kn& sin w2# + a3e~kna2t sin nsx + &c.

If t be made equal to 0, we have as the expression of the
initial state of temperatures

vx = ax sin ntx + az sin n2x + az sin n3x -f &c.

The problem consists in determining the coefficients alt a2, a3
&c., whatever be the initial state. Suppose then that we know
the values of v from x = 0 to x = X, and represent this system of
values by F(x) ; we have

F(x) = - (ax sin n^x + «2 sin njc + as sin nsx + a4 sin n^x + &C.)1. . . (e).

2.91. To determine the coefficient alt multiply both members
of the equation by x sin nx dx, and integrate from x = 0 to x = X.

The integral Ismmx sin nx dx taken between these limits is

—5 2 (— m sin nXcos mX+ n sin mJTcos wX).

m —n

If m and w are numbers chosen from the roots w1, w2> w3,

&c., which satisfy the equation - — ^= 1 — hX, we have

tan TL^\.

mX nX

tanmX t&

or m cos m X sin w X — n sin w X cos w JT = 0.

We see by this that the whole value of the integral is nothing;
but a single case exists in which the integral does not vanish,

namely, when m = n. It then becomes ^ ; and, by application of
known rules, is reduced to

--
2 4sn

1 Of the possibility of representing an arbitrary function by a series of this
form a demonstration has been given by Sir W. Thomson, Camb. Math. Journal,
Vol. m. pp. 25—27. [A, F.]

18—2

276 , THEORY OF HEAT. [CHAP. V.

It follows from this that in order to obtain the value of the
coefficient alt in equation (e), we must write

2 \x sin UjX F(x) dx — a^\X — -^~ sin Zn^X] ,

the integral being taken from x = 0 to so = X. Similarly we have
2 \x sin nzx F(x) dx=aAX—^— si

sn

In the same manner all the following coefficients may be deter
mined. It is easy to see that the definite integral 2 Ix sin nx F (x) dx

always has a determinate value, whatever the arbitrary function
F (x) may be. If the function F(x) be represented by the
variable ordinate of a line traced in any manner, the function
xF(x) sin nx corresponds to the ordinate of a second line which
can easily be constructed by means of the first. The area bounded
by the latter line between the abscissae x — 0 and x—X determines
the coefficient ait i being the index of the order of the root n.

The arbitrary function F(x) enters each coefficient under the
sign of integration, and gives to the value of v all the generality
which the problem requires; thus we arrive at the following
equation

sin n^xlx sin n%x F (x} dx

J -

sin nzx Ix sin nzx F (x) dx

- J— - e-**'« + &c.

This is the form which must be given to the general integral
of the equation

in order that it may represent the movement of heat in a solid
sphere. In fact, all the conditions of the problem are obeyed.

SECT. I.] ULTIMATE LAW OF TEMPERATURE. 277

1st, The partial differential equation is satisfied ; 2nd, the quantity
of heat which escapes at the surface accords at the same time with
the mutual action of the last layers and with the action of the air

on the surface ; that is to say, the equation -?- + hx = 0, which

each part of the value of v satisfies when x — X, holds also when
we take for v the sum of all these parts ; 3rd, the given solution
agrees with the initial state when we suppose the time nothing.

292. The roots nlt n2, 7?3, &c. of the equation

nX _, ,_

7 V" — 1 — /&-A.

tan n X.

are very unequal; whence we conclude that if the value of the
time is considerable, each term of the value of v is very small,
relatively to that which precedes it. As the time of cooling
increases, the latter parts of the value of v cease to have any
sensible influence ; and those partial and elementary states, which
at first compose the general movement, in order that the initial
state may be represented by them, disappear almost entirely, one
only excepted. In the ultimate state the temperatures of the
different layers decrease from the centre to the surface in the
same manner as in a circle the ratios of the sine to the arc
decrease as the arc increases. This law governs naturally the
distribution of heat in a solid sphere. When it begins to exist,
it exists through the whole duration of the cooling. Whatever
the function F (x) may be which represents the initial state, the
law in question tends continually to be established ; and when the
cooling has lasted some time, we may without sensible error
suppose it to exist.

293. We shall apply the general solution to the case in
which the sphere^ having been for a long time immersed in a
fluid, has acquired at all its points the same temperature. In
this case the function F(x) is 1, and the determination of the
coefficients is reduced to integrating x sin nx dx, from x = 0 to
x = X : the integral is

sin nX — nX cos n X

278 THEORY OF HEAT. [CHAP. V.

Hence the value of each coefficient is expressed thus :

2 sin n X — nX cos n X

n nX — sin nX cos n X '

the order of the coefficient is determined by that of the root n,
the equation which gives the values of n being

nX cos nX ., , v

— : TF— = 1 — h X.

sin nX

We therefore find

JiX

a— -

n n X cosec nX — cos nX '

It is easy now to form the general value which is given by the
equation

vx e~*Wl2< shifts

^ «

Denoting by et, e2, e3, &c. the roots of the equation

tan e

and supposing them arranged in order beginning with the least ;
replacing n^X, n2X, nQX} &c. by e^ e2, 63, &c., and writing instead

TT 7

of k and h their values 7^ and -^ , we have for the expression of

Ox/ xx

the variations of temperature during the cooling of a solid sphere,
which was once uniformly heated, the equation

I* C-w xV Ci ,

sm-^F
X (.

X

K e:x e1 cosec ex — cos e^

X

nn-fe

— + &c.

ea) 6 cosec 6 — cos e

Note. The problem of the sphere has been very completely discussed by
Biemann, Partielle Differentialglelchungen, §§ 61—69. [A. F.]

SECT. II.] DIFFERENT REMARKS OX THIS SOLUTION. 279

SECTION II.
Different remarks on this solution.

294<. We will now explain some of the results which may be
derived from the foregoing solution. If we suppose the coefficient
h, which measures the facility with which heat passes into the air,
to have a very small value, or that the radius X of the sphere is
very small, the least value of e becomes very small ; so that the

,

€ - h v . , , ,
equation - — — = 1 — -^ X is reduced to - = - = 1 —

e-27363

ohX
or, omitting the higher powers of e, e2 = — ^- . On the other

hand, the quantity - -- cos e becomes, on the same hypothesis,

. ex

27 Y SmX

^— And the term — is reduced to 1. On making these
K ex

X

_ 8fr t

substitutions in the general equation we have v = e Ci)X -f &c.
We may remark that the succeeding terms decrease very rapidly
in comparison with the first, since the second root n9 is very much
greater than 0 ; so that if either of the quantities h or X has
a small value, we may take, as the expression of the variations

Sht

of temperature, the equation v = e 67>j:. Thus the different
spherical envelopes of which the solid is composed retain a
common temperature during the whole of the cooling. The
temperature diminishes as the ordinate of a logarithmic curve, the
time being taken for abscissa ; the initial temperature 1 is re-

_ *hA.
duced after the time t to e CDX . In order that the initial

temperature may be reduced to the fraction — , the value of t

Y

must be ^y CD log m. Thus in spheres of the same material but

280 THEORY OF HEAT. [CHAP. V.

of different diameters, the times occupied in losing half or the
same defined part of their actual heat, when the exterior con-
ducibility is very small, are proportional to their diameters. The
same is the case with solid spheres whose radius is very small ;
and we should also "find the same result OB attributing to 'the

interior conducibility K a very great value. The statement holds

7 ~y
generally when the quantity -^ is vejy small. , We may regard

the quantity ^ as very small when the body which is being

cooled is formed of a liquid continually agitated, and enclosed in
a spherical vessel of small thickness. The hypothesis is in some
measure the same as that of perfect conducibility; the tem
perature decreases then according to the law expressed by the

Sht

equation v = e C1JX.

295. By the preceding remarks we see that in a solid sphere
which has been cooling for a long time, the temperature de
creases from the centre to the surface as the quotient of the sine
by the arc decreases from the origin where it is 1 to the end
of a given arc e, the radius of each layer being represented
by the variable length of that arc. If the sphere has a small
diameter, or if its interior conducibility is very much greater
than the exterior conducibility, the temperatures of the successive
layers differ very little from each other, since the whole arc e
which represents the radius X of the sphere is of small length.
The variation of the temperature v common to all its points

Sht

is then given by the equation v — e cux. Thus, on comparing the
respective times which two small spheres occupy in losing half
or any aliquot part of their actual heat, we find those times
to be proportional to the diameters.

_ 3M

296. The result expressed by the equation v = e CDX belongs
only to masses of similar form and small dimension. It has been
known for a long time by physicists, and it offers itself as it were
spontaneously. In fact, if any body is sufficiently small for the
temperatures at its different points to be regarded as equal, it
is easy to ascertain the law of cooling. Let 1 be the initial

SECT. II.] EXTERIOR CONDUCIBILITIES COMPARED. 281

temperature common to all points ; it is evident that the quantity
of heat which flows during the instant dt into the medium
supposed to be maintained at temperature 0 is hSvdt, denoting
by 8 the external surface of the body. On the other hand,
if C is the heat required to raise unit of weight from the tem
perature 0 to the temperature 1, we shall have CDV for the
expression of the quantity of heat which the volume V of the
body whose density is D would take from temperature 0 to

temperature 1. Hence TT/TTT- ^s tne quantity by which the

temperature v is diminished when the body loses a quantity of
heat equal to hSvdt. We ought therefore to have the equation

hSvdt gp

~' or v = e

If the form of the body is a sphere whose radius is X, we shall

-«M

have the equation v = e€DX .

297. Assuming that we observe during the cooling of the
body in question two temperatures vl and vz corresponding to
the times tt and tz, we have

hS _ log 0j — log v2
CDV~ tt-tv "'

7 Cf

We can then easily ascertain by experiment the exponent „ „,.

If the same observation be made on different bodies, and if
we know in advance the ratio of their specific heats G and C',
we can find that of their exterior conducibilities h and h'.
Reciprocally, if we have reason to regard as equal the values
h and hr of the exterior conducibilities of two different bodies,
we can ascertain the ratio of their specific heats. We see by
this that, by observing the times of cooling for different liquids
and other substances enclosed successively in the same vessel
whose thickness is small, we can determine exactly the specific
heats of those substances.

We may further remark that the coefficient K which measures
the interior conducibility does not enter into the equation

282 THEORY OF HEAT. [CHAP. V.

Thus the time of cooling iu bodies of small dimension does not
depend on the interior conducibility ; and the observation of these
times can teach us nothing about the latter property ; but it
could be determined by measuring the times of cooling in vessels
of different thicknesses.

298. What we have said above on the cooling of a sphere
of small dimension, applies to the movement of heat in a thermo
meter surrounded by air or fluid. We shall add the following
remarks on the use of these instruments.

Suppose a mercurial thermometer to be dipped into a vessel
filled with hot water, and that the vessel is being cooled freely
in air at constant temperature. It is required to find the law
of the successive falls of temperature of the thermometer.

If the temperature of the fluid were constant, and the thermo
meter dipped in it, its temperature would change, approaching
very quickly that of the fluid. Let v be the variable temperature
indicated by the thermometer, that is to say, its elevation above
the temperature of the air ; let u be the elevation of temperature
of the fluid above that of the air, and t the time corresponding
to these two values v and u. At the beginning of the instant
dt which is about to elapse, the difference of the temperature
of the thermometer from that of the fluid being v — u, the variable
v tends to diminish and will lose in the instant dt a quantity
proportional to v — u ; so that we have the equation

dv = — li (v — u) dt.

During the same instant dt the variable u tends to diminish,
and it loses a quantity proportional to u, so that we have the
equation

du = — Hudt.

The coefficient H expresses the velocity of the cooling of the
liquid in air, a quantity which may easily be discovered by ex
periment, and the coefficient h expresses the velocity with which
the thermometer cools in the liquid. The latter velocity is very
much greater than H. Similarly we may from experiment
find the coefficient h by making the thermometer cool in fluid
maintained at a constant temperature. The two equations

du = — Hudt and dv = — h (v — u) dt,

SECT. II.] ERROR OF A THERMOMETER. 283

or u — Ae~m and -j- = — hv + hAe~Ht

at

lead to the equation

v — u = le~ht + aHe~m,

a and Z> being arbitrary constants. Suppose now the initial value
of v — u to be A, that is, that the height of the thermometer
exceeds by A the true temperature of the fluid at the beginning
of the immersion; and that the initial value of u is E. We can
determine a and b, and we shall have

The quantity v — u is the error of the thermometer, that is
to say, the difference which is found between the temperature
indicated by the thermometer and the real temperature of the
fluid at the same instant. This difference is variable, and the
preceding equation informs us according to what law it tends
to decrease. We see by the expression for the difference v—u
that two of its terms containing e~u diminish very rapidly, with
the velocity which would be observed in the thermometer if it
were dipped into fluid at constant temperature. With respect
to the term which contains e~Ht, its decrease is much slower,
and is effected with the velocity of cooling of the vessel in air.
It follows from this, that after a time of no great length the
error of the thermometer is represented by the single term
HE H

e-Ht or

h-H h-H

299. Consider now what experiment teaches as to the values
of H and h. Into water at 8 '5° (octogesimal scale) we dipped
a thermometer which had first been heated, and it descended
in the water from 40 to 20 degrees in six seconds. This ex
periment was repeated carefully several times. From this we
find that the value of e~h is Q'0000421 ; if the time is reckoned
in minutes, that is to say, if the height of the thermometer be
E at the beginning of a minute, it will be #(0-000042) at the
end of the minute. Thus we find

ftlogl0 e = 4-376127l.

1 0-00004206, strictly. [A. F.]

284 THEORY OF HEAT. [CHAP. V.

At the same time a vessel of porcelain filled with water heated
to 60° was allowed to cool in air at 12°. The value of e~H in
this case was found to be 0*98514, hence that of Hlogi0e is
O006500. We see by this how small the value of the fraction
e~h is, and that after a single minute each term multiplied by
e~M is not half the ten-thousandth part of what it was at the
beginning of the minute. We need not therefore take account
of those terms in the value of v — u. The equation becomes

Hu Hu H IIu

v-u=h^n "'-"—r+a^T-

From the values found for H and A, we see that the latter
quantity h is more than 673 times greater than H, that is to
say, the thermometer cools in air more than 600 times faster

than the vessel cools in air. Thus the term -j— is certainly less

fi

than the 600th part of the elevation of temperature of the water

above that of the air, and as the term , - ^ -y— is less than

n — H fi

the 600th part of the preceding term, which is already very small,
it follows that the equation which we may employ to represent
very exactly the error of the thermometer is

Hu

V —U =

T
fl

In general if H is a quantity very great relatively to Ji, we
have always the equation

Hu

v — u = -=— .
/I

300. The investigation which we have just made furnishes
very useful results for the comparison of thermometers.

The temperature marked by a thermometer dipped into a
fluid which is cooling is always a little greater than that of the
fluid. This excess or error of the thermometer differs with the
height of the thermometer. The amount of the correction will
be found by multiplying u the actual height of the thermometer
by the ratio of H, the velocity of cooling of the vessel in air,
to h the velocity of cooling of the thermometer in the fluid. We
might suppose that the thermometer, when it was dipped into

SECT. II.] COMPARISON OF THERMOMETERS. 285

the fluid, marked a lower temperature. This is what almost
always happens, but this state cannot last, the thermometer
begins to approach to the temperature of the fluid ; at the same
time the fluid cools, so that the thermometer passes first to the
same temperature as the fluid, and it then indicates a tempera
ture very slightly different but always higher.

300*. "We see by these results that if we dip different thermo
meters into the same vessel filled with fluid which is cooling
slowly, they must all indicate very nearly the same temperature
at the same instant. Calling h, h', h", the velocities of cooling
of the thermometers in the fluid, we shall have

Hu Hu Hu

r ' IT* T~'

as their respective errors. If two thermometers are equally
sensitive, that is to say if the quantities h and Ti are the same,
their temperatures will differ equally from those of the fluid.
The values - of the coefficients h, h', h" are very great, so that the
errors of the thermometers are extremely small and often in
appreciable quantities. We conclude from this that if a thermo
meter is constructed with care and can be regarded as exact, it
will be easy to construct several other thermometers of equal
exactness. It will be sufficient to place all the thermometers
which we wish to graduate in a vessel filled with a fluid which
cools slowly, and to place in it at the same time the thermometer
which ought to serve as a model ; we shall only have to observe
all from degree to degree, or at greater intervals, and we must
mark the points where the mercury is found at the same time
in the different thermometers. These points will be at the
divisions required. We have applied this process to the con
struction of the thermometers employed in our experiments,
so that these instruments coincide always in similar circum
stances.

This comparison of thermometers during the time of cooling
not only establishes a perfect coincidence among them, and renders
them all similar to a single model ; but from it we derive also the
means of exactly dividing the tube of the principal thermometer,
by which all the others ought to be regulated. In this way we

286 THEORY OF HEAT. [CHAP. V.

satisfy the fundamental condition of the instrument, which is, that
any two intervals on the scale which include the same number of
degrees should contain the same quantity of mercury. For the
rest we omit here several details which do not directly belong to
the object of our work.

301. We have determined in the preceding articles the tem
perature v received after the lapse of a time t by an interior
spherical layer at a distance x from the centre. It is required
now to calculate the value of the mean temperature of the sphere,
or that which the solid would have if the whole quantity of heat
which it contains were equally distributed throughout the whole

mass. The volume of a sphere whose radius is x being — Q— ,

o

the quantity of heat contained in a spherical envelope whose
temperature is v, and radius x} will be vdl-^-J. Hence the
mean temperature is

PrS

J n

or

the integral being taken from x — 0 to x = X. Substitute for v
its value

— e~kniH sin n.x + — e~kn*H sin njx + — e~kn»** sin njc -f etc.

X X X

and we shall have the equation

We found formerly (Art. 293)

2 sin nt X — n,X cos n,X

a.= -- - i— .

SECT. II.] RADIUS OF SPHERE VERY GREAT. 287

We have, therefore, if we denote the mean temperature by z,

f - \o *K<iH , . N2 Kcft

= (sm 6, - ^ cos ej2 -fitx* , (sm e, - 62 cos eg) -^P „
.4 e3 26 - sin 2e * 63 2e - sin 2e

an equation in which the coefficients of the exponentials are all
positive.

302. Let us consider the case in which, all other conditions
remaining the same, the value X of the radius of the sphere
becomes infinitely great1. Taking up the construction described

r "F"

in Art. 285, we see that since the quantity —^- becomes infinite,

the straight line drawn through the origin cutting the different
branches of the curve coincides with the axis of x. We find then
for the different values of e the quantities TT, 2?r, Sir, etc.

_A !i!<
Since the term in the value of z which contains e CD x*

becomes, as the time increases, very much greater than the
following terms, the value of z after a certain time is expressed

•J--T o

without sensible error by the first term only. The index -^=

CD

KTTZ
being equal to 7^ya , we see that the final cooling is very slow

in spheres of great diameter, and that the index of e which
measures the velocity of cooling is inversely as the square of the
diameter.

303. From the foregoing remarks we can form an exact idea
of the variations to which the temperatures are subject during the
cooling of a solid sphere. The initial values of the temperatures
change successively as the heat is dissipated through the surface.
If the temperatures of the different layers are at first equal, or
if they diminish from the surface to the centre, they do not
maintain their first ratios, and in all cases the system tends more
and more towards a lasting state, which after no long delay is
sensibly attained. In this final state the temperatures decrease

1 Biemann has shewn, Part. Diff. gleich. § 69, that in the case of a very large
sphere, uniformly heated initially, the surface temperature varies ultimately as the
square root of the time inversely. [A. F.]

288 THEORY OF HEAT. [CHAP. V.

from the centre to the surface. If we represent the whole radius
of the sphere by a certain arc e less than a quarter of the
circumference, and, after dividing this arc into equal parts, take
for each point the quotient of the sine by the arc, this system of
ratios will represent that which is of itself set up among the
temperatures of layers of equal thickness. From the time when
these ultimate ratios occur they continue to exist throughout the
whole of the cooling. Each of the temperatures then diminishes
as the ordinate of a logarithmic curve, the time being taken for
abscissa. We can ascertain that this law is established by ob
serving several successive values z, z', z", z'"y etc., which denote
the mean temperature for the times t, t + ©, t + 2©, t + 3©, etc. ;

the series of these values converges always towards a geometrical

/ n

progression, and when the successive quotients -, , — , -77-, , etc.

z z z

no longer change, we conclude that the relations in question are
established between the temperatures. When the diameter of the
sphere is small, these quotients become sensibly equal as soon as
the body begins to cool. The duration of the cooling for a given
interval, that is to say the time required for the mean tem
perature z to be reduced to a definite part of itself — , increases
as the diameter of the sphere is enlarged.

304. If two spheres of the same material and different
dimensions have arrived at the final state in which whilst the
temperatures are lowered their ratios are preserved, and if we
wish to compare the durations of the same degree of cooling in
both, that is to say, the time © which the mean temperature

of the first occupies in being reduced to — , and the time © in

m

which the temperature z of the second becomes — , we must

m

consider three different cases. If the diameter of each sphere is
small, the durations © and ©' are in the same ratio as the
diameters. If the diameter of each sphere is very great, the
durations © and ©' are in the ratio of the squares of the
diameters; and if the diameters of the spheres are included
between these two limits, the ratios of the times will be greater
than that of the diameters, and less than that of their squares.

SECT. II.] EQUATION OF CONDITION HAS ONLY REAL ROOTS. 289

The exact value of the ratio has been already determined1.
The problem of the movement of heat in a sphere includes that
of the terrestrial temperatures. In order to treat of this problem
at greater length, we have made it the object of a separate
chapter8.

305. The use which has been made above of the equation
= X is founded on a geometrical construction which is very

well adapted to explain the nature of these equations. The con
struction indeed shows clearly that all the roots are real ; at the
same time it ascertains their limits, and indicates methods for
determining the numerical value of each root. The analytical
investigation of equations of this kind would give the same results.
First, we might ascertain that the equation e — X tan e = 0, in
which X is a known number less than unity, has no imaginary
root of the form m + nj—l. It is sufficient to substitute this
quantity for e ; and we see after the transformations that the first
member cannot vanish when we give to m and n real values,
unless n is nothing. It may be proved moreover that there can
be no imaginary root of any form whatever in the equation

A e cos € — X sin e
e — X tan e = 0. or = 0.

cose

In fact, 1st, the imaginary roots of the factor = 0 do not

cose

belong to the equation e — X tan e = 0, since these roots are all of

the form m + nj— 1 ; 2nd, the equation sin e — - cos e = 0 has

X

necessarily all its roots real when X is less than unity. To prove
this proposition we must consider sin e as the product of the
infinite number of factors

1 It is 9 : &=€i*X* : e^Y'2, as may be inferred from the exponent of the first
term in the expression for z, Art. 301. [A. F.]

2 The chapter referred to is not in this work. It forms part of the Suite du
inemorie sur la theorie du mouvement de la chaleur dans les corps solides. See note,
page 10.

The first memoir, entitled Theorie du mouvf.ment de la chaleur dans les corps
solides, is that which formed the basis of the Theorie analytique du mouvement de
la chaleur published in 1822, but was considerably altered and enlarged in that
work now translated. [A. F.]

F. H. 19

290 THEORY OF HEAT. {CHAP. V.

and consider cos e as derived from sin e by differentiation.

Suppose that instead of forming sin e from the product of an
infinite number of factors, we employ only the m first, and denote
the product by </>w(6)* To find the corresponding value of cose,
we take

£*.« or $„'(«).

This done, we have the equation

*.W-«*.'(«) = o.

Now, giving to the number m its successive values 1, 2, 3, 4, &a
from 1 to infinity, we ascertain by the ordinary principles of
Algebra, the nature of the functions of e which correspond to
these different values of m. We see that, whatever m the number
of factors may be, the equations in e which proceed from them
have the distinctive character of equations all of whose roots
are real. Hence we conclude rigorously that the equation

in which X is less than unity, cannot have an imaginary root1.
The same proposition could also be deduced by a different analysis
which we shall employ in one of the following chapters.

Moreover the solution we have given is not founded on the
property which the equation possesses of having all its roots
real. It would not therefore have been necessary to prove
this proposition by the principles of algebraical analysis. It
is sufficient for the accuracy of the solution that the integral
can be made to coincide with any initial state whatever; for
it follows rigorously that it must then also represent all the
subsequent states.

1 The proof given by Eiemann, Part. Diff. Gleich. § 67, is more simple. The
method of proof is in part claimed by Poisson, Bulletin de la Societe Philomatique,
Paris, 1826, p. 147. [A. F.].

. ^ w *'^<et,

rv, .

CHAPTER VI.

OF THE MOVEMENT OF HEAT IN A SOLID CYLINDER.

306. THE movement of heat in a solid cylinder of infinite
length, is represented by the equations

dv _ K (d*v ldv\ j A. T/_L ^- n
dt ~ CD (dtf + x d~x) l K V h ~dx '

which we have stated in Articles 118, 119, and 120. To inte
grate these equations we give to v the simple particular value
expressed by the equation v = ue~mt ; m being any number, and

jr

u a function of x. We denote by k the coefficient - which

enters the first equation, and by h the coefficient -^ which enters

the second equation. Substituting the value assigned to v, we
find the following condition

m dzu 1 du

7- -j-j ~ -j- —
fc axr x ctx

Next we choose for u a function of x which satisfies this
differential equation. It is easy to see that the function may
be expressed by the following series ^ 3

gx*

./ - 1 _ £ __ I

2

I X-rn

qy*

g denoting the constant -r . We shall examine more particularly

in the sequel the differential equation from which this series

19—2

292 THEORY OF HEAT. [CHAP. VI.

is derived; here we consider the function u to be known, and
we have ue~01ct as the particular value of v.

The state of the convex surface of the cylinder is subject
to a condition expressed by the definite equation

which must be satisfied when the radius x has its total value X\
whence we obtain the definite equation

oa 9a 42 ' 92 4,2 fi2 '

— 2 • V *• Tl« U

thus the number $r which enters into the particular value ue~°u
is not arbitrary. The number must necessarily satisfy the
preceding equation, which contains g and X.

We shall prove that this equation in g in which h and X
are given quantities has an infinite number of roots, and that
all these roots are real. It follows that we can give to the
variable v an infinity of particular values of the form ue~aM,
which differ only by the exponent g. We can then compose
a more general value, by adding all these particular values
multiplied by arbitrary coefficients. This integral which serves
to resolve the proposed equation in all its extent is given by
the following equation

v = a^e'^ 4- a2w2e~^w 4- «3w3e~^3< + &c.,

ffi> 9v 9a> &Ct denote all the values of g which satisfy the definite
equation ; uv uz, us, &c. denote the values of u which correspond
to these different roots; al9 az, aa, &c. are arbitrary coeffi
cients which can only be determined by the initial state of the
solid,

307. We must now examine the nature of the definite
equation which gives the values of g, and prove that all the roots
of this equation are real, an investigation which requires attentive
examination.

CHAP. VI.] THE EQUATION OF CONDITION. 293

In the series

l-*§ + |^ -^+&c.(

which expresses the value which u receives when x = X, we shall
replace *xy- by the quantity 0, and denoting this function of 0

by / (0) or y, we have

ffi /9s 0*

y =/ (0) = 1 - 0 + * "

2*. 3' 4a

the definite equation becomes

6* O3 6*

JiX ~~ 2* ~^~ 32 3* "" 2* 3* 4* ~^~

^ * ff* fj* *

1~^ + 5~s + ia 2 ~~ &C'

/'(0) denoting the function -

Each value of ^ furnishes a value for #, by means of the
equation

and we thus obtain the quantities ^, ^r2, gz, &c, which enter in
infinite number into the solution required.

The problem is then to prove that the equation

must have all its roots real. We shall J>rove in fact that the
equation f(&) — 0 has all its roots real, that the same is the
case consequently with the equation f'(0) =0, and that it follows
that the equation

'

~)

has also all its roots real, A representing the known number

hX

' 2 '

294 THEORY OF HEAT. [CHAP. VI.

308. The equation

m m

92

~*- ^ ~^~ 2 * ^

on being differentiated twice, gives the following relation

We write, as follows, this equation and all those which may
be derived from it by differentiation,

&c.,
and in general

Now if we write in the following order the algebraic equation

JT = 0, and all those which may be derived from it by differentiation,

dX d*X

and if we suppose that every real root of any one of these equa
tions on being substituted in that which precedes and in that which
follows it gives two results of opposite sign ; it is certain that the
proposed equation X = 0 has all its roots real, and that conse
quently the same is the case in all the subordinate equations

-0 &c

I — "* v« 7 t> ~"— /« i o "~~" V/j CX<V/»

dx dx* dx*

These propositions are founded on the theory of algebraic equa
tions, and have been proved long since. It is sufficient to prove
that the equations

fulfil the preceding condition. Now this follows from the general
equation

CHAP. VI.] REALITY OF THE ROOTS. 295

dly i,di+ly ^di+*u

w+v+v &+*%&-*••

di+lv
for if we give to 0 a positive value which makes the fluxion -^~i

CL\j

vanish, the other two terms -~ and -^~ receive values of opposite

sign. With respect to the negative values of 6 it is"evident, from
the nature of the function /(#), that no negative value substituted
for 6 can reduce to nothing, either that function, or any of the
others which are derived from it by differentiation: for the sub
stitution of any negative quantity gives the same sign to all the
terms. Hence we are assured that the equation y = 0 has all its
roots real and positive.

309. It follows from this that the equation / (0) = 0 or y = 0
also has all its roots real ; which is a known consequence from the
principles of algebra. Let us examine now what are the suc

cessive values which the term 6 ~hl or 0 — receives when we give

to 6 values which continually increase from 0 = 0 to 0 = GO . If a

7J

value of 6 makes y nothing, the quantity 6 — becomes nothing

«7

also ; it becomes infinite when 6 makes y nothing. Now it
follows from the theory of equations that in the case in question,
every root of y = 0 lies between two consecutive roots of y = 0,
and reciprocally. Hence denoting by #t and 03 two consecu
tive roots of the equation y' = 0, and by #2 that root of the
equation y = 0 which lies between 0l and 03, every value of 6 in
cluded between 0l and 02 gives to y a sign different from that
•which the function y would receive if 6 had a value included be

tween 02 and 03. Thus the quantity 6 — is nothing when 0=0^ it

y

is infinite when 0 = 02, and nothing when 0— 03. The quantity 0 —

y

must therefore necessarily take all possible values, from 0 to in
finity, in the interval from 0 to 0Z, and must also take all possible
values of the opposite sign, from infinity to zero, in the interval

from 02 to #3. Hence the equation A = 0 — necessarily has one

i/

296 THEORY OF HEAT. [CHAP. VI.

real root between 0X and 03 and since the equation y = 0 has all its

roots real in infinite number, it follows that the equation A — Q~

\j

has the same property. In this manner we have achieved the
proof that the definite equation

- -&c
2 '

2« 22.42 22.42.62

in which the unknown is #, has all its roots real and positive. We
A proceed to continue the investigation of the function u and of the
\ differential equation which it satisfies.

310. From the equation y -f ^| -f 6 -~ = 0, we derive the general

equation -jji + (i+ 1) J^TI + & ^r^ = 0, and if we suppose 0 = 0 we
have the equation

d^y_ 1 £y
dBi+l i + ldOif

which serves to determine the coefficients of the different terms of
the development of the function/ (0), since these coefficients depend
on the values which the differential coefficients receive when the
variable in them is made to vanish. Supposing the first term to
be known and to be equal to 1, we have the series

_ _^ __ ____ _..

If now in the equation proposed

, d*u , 1 du -
gu + -r-z+--r = Q
dor x dx

x*
we make g^ = 0, and seek for the new equation in u and 0, re

garding u as a function of 0, we find

du d?u

CHAP. VI.] SUM OF A CERTAIN SERIES. 297

Whence we conclude

_&c
* 2

It is easy to ex^es^^e^lim of this series. To obtain the

result, develope as follows the function cos_(a^siii#) in cosines of
multiple arcs. We have by known transformations ^\

o i ' \ iw*^1 -£ae-*V=l , -^ae^~l fcie-*^

2 cos (a sin x) — e * e +e e^ ,

and denoting ex ~l by o>,

aw cut)"1 aw aw"1

2 cos (a sin #) = e * e~ * + e~ a e 2 .

Developing the second member according to powers of &>, we
find the term which does not contain w in the development of
2 cos (a sin x) to be

The coefficients of a)1, «o3, a>5, &c. are nothing, the same is the case
with the coefficients of the terms which contain of1, o>~3, o>~5, &c. ;
the coefficient of aT2 is the same as that of o>2; the coefficient of o>4 is

4.6.8 22. 4. 6. 8. 10 ^

the coefficient of of4 is the same as that of &>4. It is easy to express
the law according to which the coefficients succeed ; but without
stating it, let us write 2 cos 2a? instead of (o>2 + o>~2), or 2 cos 4# in
stead of (ft)4 + &)~4), and so on : hence the quantity 2 cos (a sin x} is
easily developed in a series of the form

A + B cos 2x + Ccos 4# + D cos 6x + &c.,
and the first coefficient A is equal to

s fr °; , f t .*;.!.

if we now compare the general equation which we gave formerly
2 TT <^>(a;) = ^ l<f)(x)dx + cos # |<^(a;) cos a?(?ic + &c,

' j f ' X
- 1 4 (

298 THEORY OF HEAT. [CHAP. VI.

with the equation

2 cos (a sin x) = A 4- B cos Zx + C cos 4# + &c.,

we shall find the values of the coefficients A, B, G expressed by
definite integrals. It is sufficient here to find that of the first
coefficient A. We have then

- A = - I cos (a sin x) dx,

the integral should be taken from x = 0 to x = TT. Hence the
value of the series 1 — ^ + ^ — T* ~ w~4? — 6* + ^c> *s ^iat °^ tne

definite integral dx cos (a sin x). We should find in the same

Jo

manner by comparison of two equations the values of the successive
coefficients B, G, &c.; we have indicated these results because they
are useful in other researches which depend on the same theory.
It follows from this that the particular value of u which satisfies
the equation

d*u Idu '.If , /- . . 7
9U + j£ + ~ £c = ° 1S -J cos (« ^ sm *) fo*

the integral being taken from r = 0 to r = TT. Denoting by q this

[dx

value of u, and making u = qS, we find S = a + & — 2 > and we have

J #2

as the complete integral of the equation gu + ^—2 + - -r- = 0,

u == | a 4-6 | — T? -- >2 /cos (a; ^ sin r) Jr.
j a? in r J j

>
] Jcos (asjg sin r) dr\

a and & are arbitrary constants. If we suppose 6 = 0, we have,
as formerly,

u = I cos (x Jg sin r) dr.

With respect to this expression we add the following remarks.

^^ _ uijmjjuBjjpinwr '

' ••

311. The equation

If" /92 /9* /96

- J cos (^ sin w) c?M = 1 - ^ + ^-g - gi-pTgi + &c.

CHAP. VI.] VERIFICATION OF THE SUM. 299

verifies itself. We have in fact

Icos (0 sin 11) du = Idu (l ^ 1 , ^ \- &c.J ;

and integrating from u — 0 to u — TT, denoting by $2, S# ^6, &c.
the definite integrals

we have

Isirfudu, lsm*udu, I sin6 u du, &c.,
f fl* fi* f)6

(COS (0 Sin tt) <?M = 7T - W $2 + rj S4 - w St 4- &C.,

j

it remains to determine $2, ^4, S6, &c. The term sinn u, n being
an even number, may be developed thus

sinn u — An + Bn cos 2u + Cn cos ku + &c.

Multiplying by du and integrating between the limits u = 0 and
U = TT, we have simply I sinn u du = An7r, the other terms vanish.

From the known formula for the development of the integral
powers of sines, we have

A -- - A -! LL* A -L 4 • 5 ' 6

2 ~~ 22 ' 1 ' ~~ 24 ' 1 2 ' 6 ~ 26 ' l 2 3 ' *

Substituting these values of S# S^, S& &c., we find

1 f 6Z 6* QQ

- J cos (0 sin u) du=I-^ + ^f—p - ^ ^ ^ + &c.

We can make this result more general by taking, instead of
cos (t sin it), any function whatever (/> of t sin u.

Suppose then that we have a function <j> (z) which may be
developed thus

we shall have

* 00 = <£ + «f + f + f " + &c. ;

f t3

(f> (t sin u) = $ + £ (/>' sin w + - $' sin2 w + -5 c^"' sin3 w + &c.

X
and - |dw <f> (* sin w) = <f> + « A 6' + £ /S!2 <f>" + S #3 <#>'" -f &c. ,| (e).

7TJ 25 o

300 THEORY OF HEAT. [CHAP. VI.

Now, it is easy to see that the values of 8lt $3, $5, &c. are
nothing. With respect to $2, $4, SR) &c. their values are the
quantities which we previously denoted by A# A# AR, &c. For
this reason, substituting these values in the equation (e) we have
generally, whatever the function </> may be,

u) du

in the case in question, the function $ (z) represents cos z, and we
have (j> = 1, </>" = — 1, <£iv = 1, </>* = — 1, and so on.

312. To ascertain completely the nature of the function / (0),
and of the equation which gives the values of g, it would be
necessary to consider the form of the line whose equation is

which forms with the axis of abscissae areas alternately positive
and negative which cancel each other ; the preceding remarks, also,
on the expression of the values of series by means of definite
integrals, might be made more general. When a function of the
variable x is developed according to powers of x, it is easy to
deduce the function which would represent the same series, if the
powers x, x*, x3, &c. were replaced by cos x, cos 2aj, cos 3x, &c. By
making use of this reduction and of the process employed in the
, second paragraph of Article 235, we obtain the definite integrals
which are equivalent to given series ;' but we could not enter upon
this investigation, without departing too far from our main object.

It is sufficient to have indicated the methods which have
enabled us to express the values of series by definite integrals.

We will add only the development of the quantity 6 fj^ in a
continued fraction.

313. The undetermined y orf(0) satisfies the equation

CHAP. VI.] CORRESPONDING CONTINUED FRACTION. 301

whence we derive, denoting the functions

% tfy tfy o,

dO' W dO" '

by y\ y"> y'"> &c.,

-y =y' + 0y" or g.' =

_ __

1-2-3-4-5- &c/

&c.;

whence we conclude

Thus the value of the function — > ,x- which enters into the

7W)

definite equation, when expressed as an infinite continued
fraction, is

_0_ ±_ _0_ _0_ 6
1-2-3-4- 5-&C."

314. We shall now state the results at which we have up to |
this point arrived.

If the variable radius of the cylindrical layer be denoted by x,
and the temperature of the layer by v, a function of a? and the
time t ; the required function v must satisfy the partial differential
equation

dv _ , (d?v 1 dv
' +

for v we may assume the following value

v = ue~mt ;
u is a function of a?, which satisfies the equation

m d?u 1 du
Tw + -r-a-h- j- = 0.
K ax x ax

302 THEORY OF HEAT, [CHAP. VI.

7)1 X*

If we make 0 = — —, and consider u as a function of x, we have

K u

du Qd*u

u + d~e + 0de^°-

The following value

_i a °'2 J* J*4_ &
u — 1 — u + a — ^2 02 ~r 22 32 42

satisfies the equation in u and 0, We therefore assume the value
of u in terms of x to be

- mo? m* a? m3 x3 „

~ I 2*" + F 22 .12 ~ ¥,¥7¥7& * :c''

the sum of this series is

the integral being taken from r = 0 to r = TT. This value of v in
terms of x and m satisfies the differential equation, and retains a

finite value when x is nothing. Further, the equation hu + -j- =0

must be satisfied when x = X the radius of the cylinder. This
condition would not hold if we assigned to the quantity m any
value whatever ; we must necessarily have the equation

0

2 "1-2-3-4- 5- &c. '

i> . Vj m X*

in which 0 denotes -j- -^ .

This definite equation, which is equivalent to the following,

l fi^' °* > * \ fi ^V ^ Xr

+ 2 ~ 2~"2 + ~ *~ + 2 "" "

gives to 6 an infinity of real values denoted by 0V 0Z, 03, &c. ; the
corresponding values of m are

2 3

V2 > Y2 ' JT2 ' <"'' *

thus a particular value of v is expressed by

_22Atf?i f / x i-

Trv-e ~x*~ I cos f 2 -y, v^ sin

CHAP. VI.] FORM OF THE GENERAL SOLUTION. 303

We can write, instead of 0V one of the roots 0V 02, 03, &c., and
compose by means of them a more general value expressed by
the equation

Z-kf9i r / x \

= ale~ x* I cos f 2 -^ Jul sin qjdq

g%#3 r
A-a /

cos f 2^7^ sin c + &c.

«!, a2, a3, &c. are arbitrary coefficients : the variable q dis
appears after the integrations, which should be taken from q = 0
to q = TT.

315. To prove that this value of v satisfies all the conditions

f -, " 'WfJWM . IT- .^Sf^SJ*^ «*»***M>*iB»-

oi the problem and contains the general solution, it remains only
to determine -the coefficients alf «2, az, &c. from the initial state.
Take the equation

v = afm^u^ + a2e~mit u2 + a/rm^ u3 + &c.,

in which w1? w2, w3, &c. are the different values assumed by the
function u, or

- m xz m* x* „
~ + ~

77?

when, instead of -y-, the values ^, ^2, ^3, &c. are successively sub-

K

stituted. Making in it t = 0, wTe have the equation
V =* a^fj -f a2u2 + «3w3 + &c.,

in which F is a given function of x. Let <£ (x) be this function ;
if we represent the function ui whose index is i by >/r (xtjff^ we
have

^ (x) = a^ (a? V^) + a.^ (x Jg} + a3^ (a; v/^3) + &c.

To determine the first coefficient, multiply each member of
the equation by c^ dx, cr^ being a function of x, and integrate from
x = 0 to x = X. We then determine the function cr^ , so that after
the integrations the second member may reduce to the first tenn
only, and the coefficient al may be .found, all the other integrals

304 THEORY OF HEAT. [CHAP. VI.

having nul values. Similarly to determine the second coefficient
aa, we multiply both terms of the equation

<f> (x) = azu^ + «2w2 + o3uB -f &c.

by another factor <r2 dx, and integrate from x = 0 to x — - X. The
factor <r2 must be such that all the integrals of the second member
vanish, except one, namely that which is affected by the coefficient
* a2. In general, we employ a series of functions of x denoted by
°"i> °"2» °s» ^a wnicn correspond to the functions uiy u# us, &c. ;
each of the factors cr has the property of making all the terms
which contain definite integrals disappear in integration except
one ; in this manner we obtain the value of each of the coefficients
a,, GL, aa, &c. We must now examine what functions enjoy the

1 2 3 .^..I^IMB^B^^^^^^^':.., . J ''-...

property in question.

316. Each of the terms of the second member of the equation
is a definite integral of the form a I audx • u being a function of x
which satisfies the equation

m d?u 1 du

_

~j~ U ~T" ~7 n *l ~7~ ^

A; da? x dx

we have therefore alcrudx = -a— |(--7^-fo'-T~).
J m]\xdx dx J

Developing, by the method of integration by parts, the terms
du , d*u ,

, /V du , ~ (T C , /<r\

we have \- -^-dx = C + u |wa-

Jicau; x ) \xj

, f c£2w , -p. <&* cZcr T c?V 7

and I <7 7 o dx — V •}• -^- a — u-^ — h |w -7—, a#.

J ad? a,^ dx J dx^~

The integrals must be taken between the limits x = 0 and
x = X, by this condition we determine the quantities which enter
into the development, and are not under the integral signs. To in
dicate that we suppose x = 0 in any expression in x, we shall affect
that expression with the suffix a; and we shall give it the suffix
co to indicate the value which the function of x takes, when we
give to the variable x its last value X.

CHAP. VI.] AUXILIARY MULTIPLIERS. 305

Supposing x = 0 in the two preceding equations we have

n n , / a\ in r\ fdu da\

0 = C + [ u -} and 0 = D+-r<r — w-y-1,
\ xja \fc dxja'

thus we determine the constants C and D. Making then x = X in
the same equations, and supposing the integral to be taken from
x = 0 to x = X, we have

du,

f d?u 7 fdu da\ fdu da\ f d'2cr .

and cr -y-. ax = -7 a — u -j-\ — ( 7 a — u -y- + lu -=-5 cZa7,
J ^ \dx dxja \dx dx] a J dx2

thus we obtain the equation

-

m C -. { ( d?(r \xj] 1 fdu da a\

- -j- lo-udx = \u -r-i - — u —T— \dx + [-r- 0- — U-J-+U-)
k j J { dx dx ) \dx dx x/v

fdu dcr o-\

- (-r<T-u-j- + u-} .
\dx dx xja

p *(-

d2cr \x

317. If the quantity -^— 2 -- -r — which multiplies u under the

'

sign of integration in the second member were equal to the pro
duct of cr by a constant coefficient, the terms

u ^f-? — ) dx [ and I audx
dx j J

would be collected into one, and we should obtain for the required
integral laudx a value which would contain only determined quan
tities, with no sign of integration. It remains only to equate that
value to zero.

Suppose then the factor a to satisfy the differential equation of

,

the second order y cr + -y^ -- 4^- = 0 in the same manner as the
K cix dx

function u satisfies- the equation

m d2u 1 du

F. H. 20

306 THEORY OF HEAT. [CHAP. VI.

m and n being constant coefficients, we have

n — m[ , fdu do- <r\ fdu do- o\

—7 — \ffudx =-7-0- — u-j- +u-} — -T-O- — W-7- + M-) .
k J \dx dx x/v \dx dx x/a

Between u and a a very simple relation exists, which is dis-

covered when in the equation 7;°" + :^ -- T~~ = 0 we suPPose
cr = xs ; as the result of this substitution we have the equation

n d*s Ids _
kS*~da? + xdx~">

which shews that the function s depends on the function u given
by the equation

m d*u 1 du f.

T u + -7~2 + ~ T~ = 0-

A; cZ^2 cc rfa?
To find s it is sufficient to change m into n in the value of u ;

the value of u has been denoted by ^ (#A/ T; J , that of cr will

therefore be xty (x A/ -^ J .

We have then
cZ?^ do- a-

-j- (7 — U-j- + U-

dx dx x

= -Vf * ( Vf ) t ( VS - V^' K/l^ ( Vf )

the two last terms destroy each other, it follows that on making
x — 0, which corresponds to the suffix a, the second member
vanishes completely. We conclude from this the following equa
tion

n — m m

CHAP. VI.] VANISHING FORM. 307

It is easy to see that the second member of this equation is
always nothing when the quantities m and n are selected from
those which we formerly denoted by mv m^ m3, &c.

We have in fact

W

and hX= —.

comparing the values of /UT we see that the second member of the
equation (/) vanishes.

It follows from this that after we have multiplied by adx the
two terms of the equation

<#> (*0 = CW + aaw-a + o,w8 + &c.,

and integrated each side from a? = 0 to a; = X, in order that each of
the terms of the second member may vanish, it suffices to take

for a the quantity xu or x^r [«A-r-J .

V V K J

We must except only the case in which n = m, when the value
of laudx derived from the equation (/) is reduced to the form -,
and is determined by known rules.

318. If A / -J- = /j, and A/ T = v, we have

V A/ V A/1

If the numerator and denominator of the second member are
separately differentiated with respect to v, the factor becomes, on
making fj, = v}

We have on the other hand the equation

„ d*u 1 du « , it

A»+.-+-—»Pl or ^4--

T

20—2

308 THEORY OF HEAT. [CHAP. VI.

and also lix ^ + ^x^f = 0,

or, hty + pfy' = 0 ;
hence we have

we can therefore eliminate the quantities -\Jr' and ijr" from the
integral which is required to be evaluated, and we shall find as the
value of the integral sought

putting for /JL its value, and denoting by Ut the value which the

function u or ^rlx A / y* ) takes when we suppose x = JT. The
V V K /

index i denotes the order of the root m of the definite equa
tion which gives an infinity of values of m. If we substitute

mt or

\319. It follows from the foregoing analysis that we have the
, two equations

!x f-, fhX\*}X*U*

b = 0 and — 2-J~ 'I ' i

the first holds whenever the number i and J are different, and the
second when these numbers are equal.

Taking then the equation <j> (x) =a1ul + a2ii2 + a8ua + &c., in
which the coefficients av a2, a3, &c. are to be determined, we shall
find the coefficient denoted by a. by multiplying the two members
of the equation by xutdx, and integrating from x = 0 to x — X ;
the second member is reduced by this integration to one term
only, and we have the equation

CHAP. VI.] COMPLETE SOLUTION. 309

which gives the value of at. The coefficients al9 a2, a3, . . . ap being
thus determined, the condition relative to the initial state expressed
by the equation <f> (x) = a^ + a2u2 + a3us + &c., is fulfilled.

We can now give the complete solution of the proposed problem;
it is expressed by the following equation :

f-
J

°

i _

' '°

+ &C.

The function of a? denoted by u in the preceding equation is
expressed by

all the integrals with respect to # must be taken from a? = 0 to
x — X, and to. find the function u wer must integrate from q = 0 to
<2 = 7r; 0 (a?) is the initial value of the temperature, taken in the
interior of the cylinder at a distance # from the axis, which
function is arbitrary, and 6V 6Z, 6y &c. are the real and positive
roots of the equation

JLX-JL JL JL _L 6
"2 ~ F^ ¥^ 3 - 4^ 5-&c. '

320. If we suppose the cylinder to have been immersed for
an infinite time in a liquid maintained at a constant temperature,
the whole mass becomes equally heated, and the function (/> (x)
which represents the initial state is represented by unity. After
this substitution, the general equation represents exactly the
gradual progress of the cooling.

If t the time elapsed is infinite, the second member contains
only one term, namely, that which involves the least of all the
roots 0lt 02, 0V &c.; for this reason, supposing the roots to be
arranged according to their magnitude, and 0 to be the least, the
final state of the solid is expressed by the equation

310 THEORY OF HEAT. [CHAP. VI.

From the general solution we might deduce consequences
similar to those offered by the movement of heat in a spherical
mass. We notice first that there are an infinite number of
particular states, in each of which the ratios established between
the initial temperatures are preserved up to the end of the cooling.
I When the initial state does not coincide with one of these simple
I states, it is always composed of several of them, and the ratios of
the temperatures change continually, according as the time increases.
In general the solid arrives very soon at the state in which the
temperatures of the different layers decrease continually preserving
the same ratios. When the radius X is very small1, we find that

2ft

the temperatures decrease in proportion to the fraction e" CDX.

If on the contrary the radius X is very large2, the exponent of
e in the term which represents the final system of temperatures
contains the square of the whole radius. We see by this what
influence the dimension of the solid has upon the final velocity of
cooling. If the temperature3 of the cylinder whose radius is Xy
passes from the value A to the lesser value B, in the time T, the
temperature of a second cylinder of radius equal to X' will pass
from A to B in a different time T '. If the two sides are thin, the
ratio of the times T and Tf will be that of the diameters. If, on
I the contrary, the diameters of the cylinders are very great, the
1 ratio of the times T and T' will be that of the squares of the
diameters.

1 When X is very small, Q = -% > from tlie equation in Art. 314.^ Hence

_ &kte 2hM

e ^ becomes e, ^ .
In the text, h is the surface conducibility.

2 "When X is very large, a value of B nearly equal to one of the roots of the

B B B fi
quadratic equation 1=— — — _ will make the continued fraction in Art. 314

£i — O — 4 — O

assume its proper magnitude. Hence 0=1-446 nearly, and

_?2to0 5'78 ft*

e, x* becomes e x* .
The least root of /(0) = 0 is 1-4467, neglecting terms after 04.

3 The temperature intended is the mean temperature, which is equal to

[A. P.]

CHAPTER VII.

PROPAGATION OF HEAT IN A RECTANGULAR PRISM.

321. THE equation ^ + ^4 + j^ = 0, which we have stated

in Chapter II., Section iv., Article 125, expresses the uniform move
ment of heat in the interior of a prism of infinite length, sub
mitted at one end to a constant temperature, its initial tempera
tures being supposed nul. To integrate this equation we shall,
in the first place, investigate a particular value of v, remarking
that this function v must remain the same, when y changes sign
or when z changes sign ; an.d that its value must become infinitely
small, when the distance x is infinitely great. From this it is
easy to see that we can select as a particular value of v the
function ae~mx cos ny cos pz ; and making the substitution we find
mz — n3 — pz — 0. Substituting for n and p any quantities what
ever, we have m = Jtf+p*. The value of v must also satisfy the

definite equation Iv + 2~ = ^ when y = l or ~Z, and the equation

k V + ~dz = ° Wll6n Z = l °r ~ l (Cnapter II., Section IV., Article 125).
If we give to v the foregoing value, we have

— n sin ny + 7 cos ny = Q and — p sin pz + 7 cospz = 0,

hi hi

or -j- = pi tan pi, -r = nl tan nl.

We see by this that if we find an arc e, such that etane is equal
to the whole known quantity T I, we can take for n or p the quan-

312 THEORY OF HEAT. [CHAP. VII.

tity y. Now, it is easy to see that there are an infinite number
of arcs which, multiplied respectively by their tangents, give the
same definite product -j-, whence it follows that we can find

K

for n or p an infinite number of different values.

322. If we denote by elt e2, ea, &c. the infinite number of
arcs which satisfy the definite equation 6 tan e = ^- , we can take

for n any one of these arcs divided by I. The same would be the
case with the quantity p ; we must then take w2 = n2 + p2. If we
gave to n and p other values, we could satisfy the differential
equation, but not the condition relative to the surface. We can
then find in this manner an infinite number of particular values
of v, and as the sum of any collection of these values still satisfies
the equation, we can form a more general value of v.

Take successively for n and p all the possible values, namely,
^, -j, ^3, &c. Denoting by alf a2, a3, &c., 7>1? 62, 68, &c., con

stant coefficients, the value of v may be expressed by the following
equation :

v = (ale~x %2+%2 cos njj -f aae"a '?+^¥cos njj + &c.) \ cos n^z
4- (a^e~x ^ +n** cos n$ -f a £-* ****+"* cos njj + &c.) 52 cos n^z
+ (a^-* V^2+W32 cos n£ 4- af-****+»* cos nzy -f &c.) ba cos n3z
+ &c.

323. If we now suppose the distance x nothing, every point of
the section A must preserve a constant temperature. It is there
fore necessary that, on making x — 0, the value of v should be
always the same, whatever value we may give to y or to z ; pro
vided these values are included between 0 and I. Now, on making
x — 0, we find

v = (at cos n^y + a2 cos nzy + a3 cos n3y + &c.)
x (^ cos n^z 4- £>2 cos nzy -f &3 cos nzy + &c.).

CHAP. VII.] DETERMINATION OF THE COEFFICIENTS. 313

Denoting by 1 the constant temperature of the end A, assume
the two equations

1 = a: cos njj + «2 cos nzy + a3 cos ?? zy + &c ,
1 = \ cos n$ + b2 cos v2y + &3 cos njj + &c.

It is sufficient then to determine the coefficients alf aa, a-3, &c.,
whose number is infinite, so that the second member of the equa
tion may be always equal to unity. This problem has already
been solved in the case where the numbers nlt n3, ns, &c. form the
series of odd numbers (Chap. III., Sec. IL, Art. 177). Here
?ij, n2> n3j &c. are incommensurable quantities given by an equa
tion of infinitely high degree.

324. Writing down the equation

1 = dj cos n^y + aa cos n$ + a3 cos n.Ay + &c.,

multiply the "two members of the equation by cos n^y dy, and take
the integral from y = 0 to y — l. We thus determine the first
coefficient ar The remaining coefficients may be determined in a
similar manner.

In general, if we multiply the two members of the equation by
cos vy, and integrate it, we have corresponding to a single term
of the second member, represented by a cos nyt the integral

a Icos ny cos vy dij or ^al cos (n — v) y dy + -^ a /cos (n + v) ydy,

sin (n " ")* + ^TV sin (n +v]

and making y=-lt

a ((n 4- ii) sin (n — v)l+(n — v) sin (n -f- v)J.\

a I -~tf~?~ y

Now, every value of w satisfies the equation wtanw/ = T; the
same is the case with v, we have therefore
n tan vl = v tan z^Z ;
or n sin w£ cos vl — v sin i/£ cos ?z£ = 0.

314 THEORY OF HEAT. [CHAP. VII.

Thus the foregoing integral, which reduces to

-2 - 2 (n sm nl cos vl—v cos nl sin vl),

is nothing, except only in the case where n — v. Taking then the
integral

a jsin (n — v)l sin (n + v) I]

2 [ n-v n + v J '

we see that if we have n = v, it is equal to the quantity

sin 2

It follows from this that if in the equation

1 = ai cos 71$ + «2 cos n2y + as cos nzy + &c.

we wish to determine the coefficient of a term of the second
member denoted by a cos nyy we must multiply the two members
by cos ny dy, and integrate from y = 0 to y — L We have the
resulting equation

fl * * A sin2nZ\ 1 .

cos nydy = -^a\l H — - — 1 = - sin nl,
Jo y J 2 V 2» / fi

whence we deduce x— ^ - . _ 7 = - a. In this manner the coeffi-
2nl + sin 2nl 4

cients a^ a2, a3, &c. may be determined ; the same is the case
with blt 62, 63, &c., which are respectively the same as the former

coefficients.

325. It is easy now to form the general value of v. 1st, it

d?v dzv d?v

satisfies the equation -Y-.+ T-^ + -T^ = O; 2nd. it satisfies the two
dx dy dz

conditions k-j- + hv = 0, and Jc-j- + hv — 0; 3rd, it gives a constant

value to v when we make x — 0, whatever else the values of y and
z may be, included between 0 and Z; hence it is the complete
solution of the proposed problem.

We have thus arrived at the equation

cos n^y sin nj, cos nzy sin nsl cos nzy „
in 2 2n7 + sin 2 2w^ + sin2w« C'J

1 _ sin n cos n
~

CHAP. VII.] THE SOLUTION. 315

or denoting by 61} e2, e3, &c. the arcs nj., ntl, n3l, &c.

e.y . e9y . e»y

sin e, cos -~ sin e2 cos -~- sin e3 cos -y-

1

+ _ + &c.

4 2ex + sin el 2ea + sin e2 2e3 + sin e3

an equation which holds for all values of y included between
0 and I, and consequently for all those which are included between
0 and — I, when x = 0.

Substituting the known values of al9 blt aa, &2, aa, b3, &c. in
the general value of v, we have the following equation, which
contains the solution of the proposed problem,

v _ smnjcosnf fsmnjcoan.y y^~^ ,
4.4 2

sin njcosnjs / sin n^cosn.y v^TT^ , &c
* in 2?i2Z V 2^? + sin 2n^

sin w ? cos n.z f sin w.Z cos n. y

j __ s _ § _ _ I _ i'~ g— a

2/i3^ + sin 2n2l \ZriJ + sin 2/i^
+ &c .................................................... (E).

The quantities denoted by nlt n^ nB, &c. are infinite in
number, and respectively equal to the quantities j , j , ,3 , &c. ;

the arcs, e1, e2, eg, &c., are the roots of the definite equation

hi
e tan e = -=- .

326. The solution expressed by the foregoing equation E is
the only solution which belongs to the problem ; it represents the

general integral of the equation -^ + -^— 2 + y-2 = 0, in which the

arbitrary functions have been determined from the given condi
tions. It is easy to see that there can be no different solution.
In fact, let us denote by -fy (as, y, z] the value of v derived from the
equation (E), it is evident that if we gave to the solid initial tem
peratures expressed by ty(x, y, z), no change could happen in the
system of temperatures, provided that the section at the origin
were retained at the constant temperature 1: for the equation

j-a + -5-5 + ~J~> — 0 being satisfied, the instantaneous variation of
dx dy dz"

31 G THEORY OF HEAT. [CHAP. VII.

the temperature is necessarily nothing. The same would not be
the case, if after having given to each point within the solid whose
co-ordinates are x, yt z the initial temperature ty(x, y, z), we gave
to all points of the section at the origin the temperature 0. We
see clearly, and without calculation, that in the latter case the
state of the solid would change continually, and that the original
heat which it contains would be dissipated little by little into the
air, and into the cold mass which maintains the end at the tem
perature 0. This result depends on the form of the . function
ty(x, y, z), which becomes nothing when x has an infinite value as
the problem supposes.

A similar effect would exist if the initial temperatures instead
of being + ty (x, y, z) were — -^ (#, y, z] at all the internal points
of the prism ; provided the section at the origin be maintained
always at the temperature 0. In each case, the initial tempera
tures would continually approach the constant temperature of the
medium, which is 0 ; and the final temperatures would all be nul.

327. These preliminaries arranged, consider the movement of
heat in two prisms exactly equal to that which was the subject of
the problem. For the first solid suppose the initial temperatures
to be + ^(a?, y, s), and that the section at origin A is maintained
at the fixed temperature 1. For the second solid suppose the
initial temperatures to be — ^ (x, y, z), and that at the origin A
all points of the section are maintained at the temperature 0. It
is evident that in the first prism the system of temperatures can
not change, and that in the second this system varies continually
up to that at which all the temperatures become nul.

If now we make the two different states coincide in the same
solid, the movement of heat is effected freely, as if each system
alone existed. In the initial state formed of the two united
systems, each point of the solid has zero temperature, except the
points of the section A, in accordance with the hypothesis. Now
the temperatures of the second system change more and more,
and vanish entirely, whilst those of the first remain unchanged.
Hence after an infinite time, the permanent system of tempera
tures becomes that represented by equation E, or v = ^r(#, y, z].
It must be remarked that this result depends on the condition
relative to the initial state ; it occurs whenever the initial heat

CHAP. VII.]

GEOMETRICAL CONSTRUCTION.

31'

contained in the prism is so distributed, that it would vanish
entirely, if the end A were maintained at the temperature 0.

328. We may add several remarks to the preceding solution.
1st, it is easy to see the nature of the equation e tan e = -j- ; we

need only suppose (see fig. 15) that we have constructed the curve
u = e tan e, the arc e being taken for abscissa, and u for ordinate.
The curve consists of asymptotic branches.

Fig. 15.

The abscissa? which correspond to the asymptotes are ^TT,

357

o71"' o77"' 971"' &c-> those which correspond to points of intersec
tion are TT, 2?r, 3?r, &c. If now we raise at the origin an ordinate
equal to the known quantity ~r , and through its extremity draw

K.

a parallel to the axis of abscissa?, the points of intersection will
give the roots of the proposed equation e tan e = -j- . The con
struction indicates the limits between which each root lies. We
shall not stop to indicate the process of calculation which must be
employed to determine the values of the roots. Researches of
this kind present no difficulty.

329. 2nd. We easily conclude from the general equation (E)
that the greater the value of x becomes, the greater that term of

the value of v becomes, in which we find the fraction jT "1*"*'*l%
with respect to each of the following terms. In fact, nl9 nz, w3,
&c. being increasing positive quantities, the fraction e~'rx 2nr is

318 THEORY OF HEAT. [CHAP. VII.

greater than any of the analogous fractions which enter into the
subsequent terms.

Suppose now that we can observe the temperature of a point
on the axis of the prism situated at a very great distance x, and
the temperature of a point on this axis situated at the distance
x + 1, 1 being the unit of measure ; we have then y — 0, z = 0,
and the ratio of the second temperature to the first is sensibly
equal to the fraction e~ ^2ni\ This value of the ratio of the tem
peratures at the two points on the axis becomes more exact as the
distance x increases.

It follows from this that if we mark on the axis points each of
which is at a distance equal to the unit of measure from the pre
ceding, the ratio of the temperature of a point to that of the point
which precedes it, converges continually to the fraction e~^2ni'z;
thus the temperatures of points situated at equal distances end
by decreasing in geometrical progression. This law always holds,
whatever be the thickness of the bar, provided we consider points
situated at a great distance from the source of heat.

It is easy to see, by means of the construction, that if the
quantity called I, which is half the thickness of the prism, is very
small, n{ has a value very much smaller than nz, or ??3, &c. ; it
follows from this that the first fraction e~x^2ni* is very much
greater than any of the analogous fractions. Thus, in the case in
which the thickness of the bar is very small, it is unnecessary to
be very far distant from the source of heat, in order that the tem
peratures of points equally distant may decrease in geometrical
progression. The law holds through the whole extent of the bar.

330. If the half thickness Z is a very small quantity, the
general value of v is reduced to the first term which contains
e-x\/zn^^ Thus the function v which expresses the temperature of
a point whose co-ordinates are x, y, and z, is given in this case by
the equation

(4 sin nl \2 -x-Jzn?

=—, . . 7 cos ny cos nz e ,

2nl + sm 2nlJ

the arc e or nl becomes very small, as we see by the construction.
The equation e tan e = j- reduces then to e2 = -r ; the first value of

CHAP. VII.] CASE OF A THIN BAR. 319

e, or elf is \J j- ; by inspection of the figure we know the values of
the other roots, so that the quantities elt e2, e8, e4, e6, &c. are the
following A/ -j- , TT, 27r, STT, 4-Tr, &c. The values of nv nv n3, n^ ny &c.
are, therefore,

!_ /h 7T 27T 3?T

v^v &' J ' i ' ~i '

whence we conclude, as was said above, that if I is a very small
quantity, the first value n is incomparably greater than all the
others, and that we must omit from the general value of v all the
terms which follow the first. If now we substitute in the first
term the value found for n, remarking that the arcs nl and 2nl are
equal to their sines, we have

hl\ x /?»

the factor A/ -j- which enters under the symbol cosine being very

small, it follows that the temperature varies very little, for
different points of the same section, when the half thickness I is
very small. This result is so to speak self-evident, but it is useful
to remark how it is explained by analysis. The general solution
reduces in fact to a single term, by reason of the thinness of the
bar, and we have on replacing by unity the cosines of very small

A*

arcs v = e~x*kl , an equation which expresses the stationary tempe

ratures in the case in question.

We found the same equation formerly in Article 76 ; it is
obtained here by an entirely different analysis.

331. The foregoing solution indicates the character of the
movement of heat in the interior of the solid. It is easy to see
that when the prism has acquired at all its points the stationary
temperatures which we are considering, a constant flow of heat
passes through each section perpendicular to the axis towards the
end which was not heated. To determine the quantity of flow
which corresponds to an abscissa x, we must consider that the
quantity which flows during unit of time, across one element of

320 THEORY OF HEAT. [CHAP. VII.

the section, is equal to the product of the coefficient k, of the area

£/75

dydzj of the element dt, and of the ratio -=- taken with the nega
tive sign. We must therefore take the integral — k I dy I dz -=— ,

from z = 0 to z — I, the half thickness of the bar, and then from
y = 0 to y = I. We thus have the fourth part of the whole flow.

The result of this calculation discloses the law according to
which the quantity of heat which crosses a section of the bar
decreases ; and we see that the distant parts receive very little
heat from the source, since that which emanates directly from it
is directed partly towards the surface to be dissipated into the air.
That which crosses any section whatever of the prism forms, if we
may so say, a sheet of heat whose density varies from one point
of the section to another. It is continually employed to replace
the heat which escapes at the surface, through the whole end of
the prism situated to the right of the section : it follows therefore
that the whole heat which escapes during a certain time from this
part of the prism is exactly compensated by that which penetrates
it by virtue of the interior conducibility of the solid.

To verify this result, we must calculate the produce of the flow
established at the surface. The element of surface is dxdy, and v
being its temperature, hvdxdy is the quantity of heat which
escapes from this element during the unit of time. Hence the

integral h\dx\dyv expresses the whole heat which has escaped

from a finite portion of the surface. We must now employ the
known value of v in yt supposing z = 1, then integrate once from
y = QiQy = l, and a second time from x = x up to x = oo . We
thus find half the heat which escapes from the upper surface of
the prism ; and taking four times the result, we have the heat lost
through the upper and lower surfaces.

If we now make use of the expression h Ida) I dz v, and give to

y in v its value I, and integrate once from z = 0 to z = l, and a
second time from x = 0 to x = oo ; we have one quarter of the heat
which escapes at the lateral surfaces.

The integral /? I dx \dy v, taken between the limits indicated gives

CHAP. VII.] HEAT LOST AND TRANSMITTED. 321

— . sin ml cos nl e~x^mi+n*,

and the integral h I dx Idz v gives
a

cos ml sin

.
n v m2 + n

Hence the quantity of heat which the prism loses at its surface,
throughout the part situated to the right of the section whose
abscissa is x, is composed of terms all analogous to

sin ml cos nl + - cos ml sin

in nl\ .
}

On the other hand the quantity of heat which during the same
time penetrates the section whose abscissa is x is composed of
terms analoous to

'sin mlsiD.nl ;
mn

the following equation must therefore necessarily hold

— sin ml sin nl = . sin ml cos nl

H cos ml sin nl,

or k (mz + ?i2) sin ml sin nl = hm cos mZ sin nl + hn sin ml cos wZ ;
now we have separately,

km? sin ml cos wZ = ^/?i cos ml sin wZ,

m sin ml h

or — i- = 7 5

cos mZ k

we have also

A;?i2 sin nl sin mZ = hn cos nZ sin mZ,

n sin ??Z A

or — r = 7 .

cos ?iZ k

Hence the equation is satisfied. This compensation which is in
cessantly established between the heat dissipated and the heat
transmitted, is a manifest consequence of the hypothesis ; and
analysis reproduces here the condition which has already been ex-

F. H. 21

322 THEORY OF HEAT. [CHAP. VII.

pressed; but it was useful to notice this conformity in a new
problem, which had not yet been submitted to analysis.

332. Suppose the half side I of the. square which serves as the
base of the prism to be very long, and that we wish to ascertain the
law according to which the temperatures at the different points of
the axis decrease ; we must give to y and z mil values in the
general equation, and to I a very great value. Now the construc

tion shews in this case that the first value of e is -^ , the second

-x- , the third — , &c. Let us make these substitutions in the general
2 2i

equation, and replace n^ nj, nal, nj, &c. by their values Q,-~-,

A 2t

f> t-r X IT

-£-, ~ } and also substitute the fraction a for e"1 * ; we then find

L L

-&C.

We see by this result that the temperature at different points
of the axis decreases rapidly according as their distance from the
origin increases. If then we placed on a support heated and
maintained at a permanent temperature, a prism of infinite height,
having as base a square whose half side I is very great; heat would
be propagated through the interior of the prism, and would be dis
sipated at the surface into the surrounding air which is supposed
to be at temperature 0. When the solid had arrived at a fixed
state, the points of the axis would have very unequal tempera-
tares, and at a height equal to half the side of the base the
temperature of the hottest point would be less than one fifth part
of the temperature of the base.

CHAPTER VIII.

OF THE MOVEMENT OF HEAT IN A SOLID CUBE.

333. IT still remains for us to make use of the equation
dv K /d?v d*v a

which represents the movement of heat in a solid cube exposed
to the action .of the air (Chapter II., Section v.). Assuming, in
the first place, for v the very simple value e~mtcosnx cospycosqz,
if we substitute it in the proposed equation, we have the equa
tion of condition m = k (n* + p* + q*), the letter k denoting the

TT-

coefficient . It follows from this that if we substitute for

n, p, q any quantities whatever, and take for m the quantity
k(nz + p* + q2), the preceding value of v will always satisfy the
partial differential equation. We have therefore the equation
v = e- k (n*+P* +q2)t cos nx cospycosqz. The nature of the problem
requires also that if x changes sign, and if y and z remain the
same, the function should not change ; and that this should also
hold with respect to y or z: now the value of v evidently satisfies
these conditions.

334. To express the state of the surface, we must employ the
following equations :

.(6).

21—2

324* THEORY OF HEAT. [CHAP. VIII.

These ought to be satisfied when x — ± a, or y — ± a, or g — ± a,
The centre of the cube is taken to be the origin of co-ordinates :
and the side is denoted by a.

The first of the equations (6) gives

+ e"mt n sin nx cospy cos qz + -^ cos nx cospy cos qz = 0,

or + n tan nx + ^=0,

K

an equation which must hold when x = ± a.

It follows from this that we cannot take any value what
ever for nt but that this quantity must satisfy the condition

nata>una — -^a. We must therefore solve the definite equation
J\.

e tan e = -^a, which gives the value of e, and take n = - . Now the
J\. &

equation in e has an infinity of real roots ; hence we can find for
n an infinity of different values. We can ascertain in the same
manner the values which may be given to p and to q ; they are
all represented by the construction which was employed in the
preceding problem (Art. 321). Denoting these roots by n^n^n^ &c.;
we can then give to v the particular value expressed by the
equation

cos z

provided we substitute for n one of the roots nv nz, n3, &c., and
select p and q in the same manner.

335. We can thus form an infinity of particular values of v,
and it evident that the sum of several of these values will also
satisfy the differential equation (a), and the definite equations (£).
In order to give to v the general form which the problem requires,
we may unite an indefinite number of terms similar to the term

cos nx wspy cos qz.
The value of v may be expressed by the following equation :
v = (at cos n^x e~kn& + a2 cos nzx e~kn^ + a3 cos n3x e~ *"'& + &c.),
(bl cos n^y Q-I™M + ^ cos n^ e-kn?t _j_ £3 COs n$ e~kn*H + &c.),
(Cj cos n^z e~kn^ + c2 cos n2z er*"**' + c8 cos nsy e~kn»H + &c.).

CHAP. VIII.] GENERAL VALUE OF V. 325

The second member is formed of the product of the three
factors written in the three horizontal lines, and the quantities
ax , a2, «3, &c. are unknown coefficients. Now, according to the
hypothesis, if t be made = 0, the temperature must be the same at
all points of the cube. We must therefore determine a1} a2, a3, &c.,
so that the value of v may be constant, whatever be the values of
x, y, and z, provided that each of these values is included between
a and — a. Denoting by 1 the initial temperature at all points of
the solid, we shall write down the equations (Art. 323)

1 = a: cos n^x + a2 cos nzx + aa cos nsx + &c.,
1 = &x cos nty + 6a cos n2y + b3 cos n^y + &c.,
1 = cl cos n^z + ca cos nzz + ca cos nBz + &c.,

in which it is required to determine alt at, as, &c. After multi
plying each member of the first equation by cosnx, integrate
from # = 0 to X — CL-. it follows then from the analysis formerly
employed (Art. 324) that we have the equation

sin n^a cos n^x sin n^a cos n^x sin nza cos njc

1 = i : ^T? r -f i : ^-s r + , : gin £

tn^\
nja, )

+ &c.
Denoting by ^ the quantity ^ f 1 H — ' * j, we have

_ . sin n.a « sin n.a sin n.a p

1 = — cos njc -\ — cos n^x H ^— cos nsx -f &c.

This equation holds always when we give to x a value included
between a and — a,

From it we conclude the general value of v, which is given by
the following equation

/sin n. a ,.„ 2/ sin n a , ,. „ \

v = ( L cos n^x e~kni t -f cos njc e~kn* f + &c. ) ,

(s- — i- cos njje~kniH ^ — cos n$ e~ina^ + &c.J,

/sin n CL , ,, sin nna

— ros M z fi ~ kn* f -I si cos n^z e '

326 THEORY OF HEAT. [CHAP. VIII.

336. The expression for v is therefore formed of three similar
functions, one of x, the other of y, and the third of z, which is
easily verified directly.

In fact, if in the equation

dt~

we suppose v — XYZ\ denoting by X a function of x and t,
by Y a function of y and t, and by Z a function of z and t, we have

_ . ,

" + +" *W - F **-z&)

i ax i dY i dz

x"^ + rW+^^

which implies the three separate equations

~dt ~ d^ ' di dy" dt~ dz '
We must also have as conditions relative to the surface,

dV k V n
^+^F==°

whence we deduce

=,=,.

dx K dy K dz K

It follows from this, that, to solve the problem completely, it is

// ?/ ri ?/

enough to take the equation -^ = k -^ , and to add to it the

equation of condition -p + ^u — 0, which must hold when x = a.

We must then put in the place of a?, either T/ or #, and we shall
have the three functions X} Y} Z, whose product is the general
value of v.

Thus the problem proposed is solved as follows :

, 0 ;

cos

CHAP. VIII.] ONE SOLUTION ONLY. 327

nl} w2, ?ig, &c. being given by the following equation

ha

in which e represents na and the value of /x, is

2 V 2n^a } '

In the same manner the functions <f> (yy t), $ (z, t) are found.

337. We may be assured that this value of v solves the pro
blem in all its extent, and that the complete integral of the partial
differential equation (a) must necessarily take this form in order
to express the variable temperatures of the solid.

In fact, the expression for v satisfies the equation (a) and the
conditions relative to the surface. Hence the variations of tempe
rature which result in one instant from the action of the molecules
and from the "action of the air on the surface, are those which we
should find by differentiating the value of v with respect to the
time t. It follows that if, at the beginning of any instant, the
function v represents the system of temperatures, it will still
represent those which hold at the commencement of the following
instant, and it may be proved in the same manner that the vari
able state of the solid is always expressed by the function v, in
which the value of t continually increases. Now this function
agrees with the initial state: hence it represents all the later
states of the solid. Thus it is certain that any solution which
gives for v a function different from the preceding must be wrong.

338. If we suppose the time t, which has elapsed, to have
become very great, we no longer have to consider any but the
first term of the expression for v ; for the values nv n^ n3, &c. are
arranged in order beginning with the least. This term is given
by the equation

/sin ??1 a\5
v = (— -) cos n^x cos n^y cos n^z

this then is the principal state towards which the system of tem
peratures continually tends, and with which it coincides without
sensible error after a certain value of t. In this state the tempe-

328 THEORY OF HEAT. [CHAP. VIII.

rature at every point decreases proportionally to the powers of
the fraction e~skn^-} the successive states are then all similar, or
rather they differ only in the magnitudes of the temperatures
which all diminish as the terms of a geometrical progression, pre
serving their ratios. We may easily find, by means of the pre
ceding equation, the law by which the temperatures decrease from
one point to another in direction of the diagonals or the edges of
the cube, or lastly of a line given in position. We might ascer
tain also what is the nature of the surfaces which determine the
layers of the same temperature. We see that in the final and
regular state which we are here considering, points of the same
layer preserve always equal temperatures, which would not hold
in the initial state and in those which immediately follow it.
During the infinite continuance of the ultimate state the mass is
divided into an infinity of layers all of whose points have a com
mon temperature.

339. It is easy to determine for a given instant the mean
temperature of the mass, that is to say, that which is obtained by
taking the sum of the products of the volume of each molecule
by its temperature, and dividing this sum by the whole volume.

We thus form the expression 1 1 1 3% — , which is that of the

mean temperature V. The integral must be taken successively
with respect to x, y, and z, between the limits a and — a : v being
equal to the product X YZ} we have

thus the mean temperature is fl-gpl > since the three complete
integrals have a common value, hence

e-^+ Ac.

V nfl J PI \ nta

The quantity na is equal to e, a root of the equation e tan e = -~ ,

and //, is equal to x (l + — 5— J • We have then, denoting the
different roots of this equation by 61} ea, e8, &c.,

CHAP. VIII.] CUBE AND SPHERE COMPARED. 329

6, is between 0 and - TT, e2 is between TT and — , e3 between 2?r and

- TT, the roots e2, 6g, e4, &c. approach more and more nearly to the

inferior limits TT, 2-Tr, 3-7T, &c., and end by coinciding with them
when the index i is very great. The double arcs 2el5 2e2, 2e3, &c.,
are included between 0 and TT, between 2?r and 3?r, between 4?r
and OTT ; for which reason the sines of these arcs are all positive :

. . sin 2e, .. , sin 2e p . . .

the quantities 1 H — - — - , 1 H — ^ — -2 , &c., are positive and included

161 ^€2

between 1 and 2. It follows from this that all the terms which
enter into the value of ^ V are positive.

340. We propose now to compare the velocity of cooling in
the cube, with that which we have found for a spherical mass.
We have seen that for either of these bodies, the system of tem
peratures converges to a permanent state which is sensibly attained
after a certain time ; the temperatures at the different points of
the cube then diminish all together preserving the same ratios,
and the temperatures of one of these points decrease as the terms
of a geometric progression whose ratio is not the same in the two
bodies. It follows from the two solutions that the ratio for the

. 3 —3— Je

sphere is e~ 'n and for the cube e «2 . The quantity n is given by
the equation

cos na h

na - - = 1 — ^,<7,

sm na K

a being the semi-diameter of the sphere, and the quantity e is given
by the equation e tan e = -^a, a being the half side of the cube.

This arranged, let us consider two different cases; that in
which the radius of the sphere and the half side of the cube are
each equal to a, a very small quantity ; and that in which the
value of a is very great. Suppose then that the two bodies are of

330 THEORY OF HEAT. [CHAP. VIII.

small dimensions; -^having a very small value, the same is the
case with e, we have therefore -^ = e2, hence the fraction

-3-Jfe - —

e <*<* is equal to e cva .

Thus the ultimate temperatures which we observe are expressed in

_!^ TP . ,, . na cos na h

the form Ae CDa. If now in the equation — : — - =1 — -j^a, we

sin na K.

suppose the second member to differ very little from unity, we find

^ n*a i ^ £ A- -W - -•—-

-^= -^-, hence the fraction e is e cva.
JK. o

We conclude from this that if the radius of the sphere is very-
small, the final velocities of cooling are the same in that solid and
in the circumscribed cube, and that each is in inverse ratio of the
radius ; that is to say, if the temperature of a cube whose half side
is a passes from the value A to the value B in the time t, a sphere
whose semi-diameter is a will also pass from the temperature A
to the temperature B in the same time. If the quantity a were
changed for each body so as to become a, the time required for
the passage from A to B would have another value t', and the
ratio of the times t and t' would be that of the half sides a and a.
The same would not be the case when the radius a is very great :
for 6 is then equal to JTT, and the values of na are the quantities
TT, 27T, 3-7T, 4?r, &c.

We may then easily find, in this case, the values of the frac
tions e & , e'^2 ; they are e~^ and e~~"* .

From this we may derive two remarkable consequences: 1st, when
two cubes are of great dimensions, and a and a are their half-
sides ; if the first occupies a time t in passing from the temperature
A to the temperature B, and the second the time t' for the same
interval ; the times t and t' will be proportional to the squares a2
and az of the half-sides. We found a similar result for spheres of
great dimensions. 2nd, If the length a of the half-side of a cube
is considerable, and a sphere has the same magnitude a for radius,
and during the time t the temperature of the cube falls from A to
B} a different time t' will elapse whilst the temperature of the

CHAP. VIII.] REMARKS. 331

sphere is falling from A to JB, and the times t and t' are in the
ratio of 4 to 3.

Thus the cube and the inscribed sphere cool equally quickly
when their dimension is small ; and in this case the duration of
the cooling is for each body proportional to its thickness. If the
dimension of the cube and the inscribed sphere is great, the final
duration of the cooling is not the same for the two solids. This
duration is greater for the cube than for the sphere, in the ratio of
4 to 3, and for each of the two bodies severally the duration of the
cooling increases as the square of the diameter.

341. We have supposed the body to be cooling slowly in at
mospheric air whose temperature is constant. We might submit
the surface to any other condition, and imagine, for example, that
all its points preserve, by virtue of some external cause, the fixed
temperature 0. The quantities n, p, q, which enter into the value
of v under the symbol cosine, must in this case be such that cos nx
becomes nothing when x has its complete value a, and that the
same is the case with cos py and cos qz. If 2a the side of the
cube is represented by TT, 2?r being the length of the circumference
whose radius is 1 ; we can express a particular value of v by the
following equation, which satisfies at the same time the general
equation of movement of heat, and the state of the surface,

..»

v = e cb cos x . cos y . cos z.

This function is nothing, whatever be the time tt when x or y or z

receive their extreme values + - or — - : but the expression for the

2i 2*

temperature cannot have this simple form until after a consider
able time has elapsed, unless the given initial state is itself
represented by cos x cos y cos z. This is what we have supposed
in Art. 100, Sect. Yin. Chap. I. The foregoing analysis proves the
truth of the equation employed in the Article we have j ust cited.

Up to this point we have discussed the fundamental problems
in the theory of heat, and have considered the action of that
element in the principal bodies. Problems of such kind and order
have been chosen, that each presents a new difficulty of a higher
degree. We have designedly omitted a numerous variety of

332 THEORY OF HEAT. [CHAP. VIII.

intermediate problems, such as the problem of the linear movement
of heat in a prism whose ends are maintained at fixed temperatures,
or exposed to the atmospheric air. The expression for the varied
movement of heat in a cube or rectangular prism which is cooling
in an aeriform medium might be generalised, and any initial
state whatever supposed. These investigations require no other
principles than those which have been explained in this work,

A memoir was published by M. Fourier in the Memoir es de V Academic des
Sciences, Tome vii. Paris, 1827, pp. 605 — 624, entitled, Memoire sur la distinction des
racines imaginaires, et sur Vapplication des theoremes d'analyse algebrique aux
equations transcendantes qui dependent de la theorie de la chaleur. It contains a
proof of two propositions in the theory of heat. If there be two solid bodies of
similar convex forms, such that corresponding elements have the same density,
specific capacity for heat, and conductivity, and the same initial distribution of
temperature, the condition of the two bodies will always be the same after times
which are as the squares of the dimensions, when, 1st, corresponding elements
of the surfaces are maintained at constant temperatures, or 2nd, when the tem
peratures of the exterior medium at corresponding points of the surface remain
constant.

For the velocities of flow along lines of flow across the terminal areas *, s' of
corresponding prismatic elements are as u-v : u' -v', where (u, v), (i/, 1/) are tem
peratures at pairs of points at the same distance £ A on opposite sides of s and s';
and if n : n' is the ratio of the dimensions, u-v : u'-v'=n':n. If then, dt, dt' be
corresponding times, 'the quantities of heat received by the prismatic elements are
as sk (u -v) dt : s'k (u' - i/) dtf, or as n^n'dt : itf'ndt'. But the volumes being as
n3 : n'3, if the corresponding changes of temperature are always equal we must have

n?n'dt _ n'2ndt' dt__<n?_
ri* :; ra'3 '°r<^"'~^*

In the second case we must suppose H : H'=ri: n. [A. F.]

CHAPTER IX.

OF THE DIFFUSION OF HEAT.

FIRST SECTION.

Of the free movement of heat in an infinite line.

342. HERE we consider the movement of heat in a solid
homogeneous mass, all of whose dimensions are infinite. The
solid is divided by planes infinitely near and perpendicular to a
common axis ; and it is first supposed that one part only of the
solid has been heated, that, namely, which is enclosed between
two parallel planes A and B, whose distance is g ; all other parts
have the initial temperature 0 ; but any plane included between
A and B has a given initial temperature, regarded as arbitrary,
and common to every point of the plane ; the temperature is dif
ferent for different planes. The initial state of the mass being
thus defined, it is required to determine by analysis all the suc
ceeding states. The movement in question is simply linear, and
in direction of the axis of the plane ; for it is evident that there
can be no transfer of heat in any plane perpendicular to the axis,
since the initial temperature at every point in the plane is the
same.

Instead of the infinite solid we may suppose a prism of very
small thickness, whose lateral surface is wholly impenetrable to
heat. The movement is then considered only in the infinite line
which is the common axis of all the sectional planes of the prism.

The problem is more general, when we attribute temperatures
entirely arbitrary to all points of the part of the solid which has

334 THEORY OF HEAT. [CHAP. IX.

been heated, all other points of the solid having the initial tem
perature 0. The laws of the distribution of heat in an infinite
solid mass ought to have a simple and remarkable character ;
since the movement is not disturbed by the obstacle of surfaces,
or by the action of a medium.

343. The position of each point being referred to three rect
angular axes, on which we measure the co-ordinates x, y, z, the
temperature sought is a function of the variables x, y, z, and of
the time t. This function v or <£ (x, y, z, t) satisfies the general
equation

dv _ K fdzv d*v dzv\ , .

dt~ C7)(dx2+d^ + dz2) '

Further, it must necessarily represent the initial state which is
arbitrary; thus, denoting by F(x, y, z) the given value of the
temperature at any point, taken when the time is nothing, that is
to say, at the moment when the diffusion begins, we must have

<£(*, y, z, 0) = F(x, y, z) (5).

Hence we must find a function v of the four variables x, y, z, t,
which satisfies the differential equation (a) and the definite equa
tion (&).

In the problems which we previously discussed, the integral is
subject to a third condition which depends on the state of the
surface : for which reason the analysis is more complex, and the
solution requires the employment of exponential terms. The
form of the integral is very much more simple, when it need only
satisfy the initial state; and it would be easy to determine at
once the movement of heat in three dimensions. But in order to
explain this part of the theory, and to ascertain according to what
law the diffusion is effected, it is preferable to consider first the
linear movement, resolving it into the two following problems : we
shall see in the sequel how they are applied to the case of three
dimensions.

344. First problem : a part a b of an infinite line is raised at
all points to the temperature 1 ; the other points of the line are at
the actual temperature 0 ; it is assumed that the heat cannot be
dispersed into the surrounding medium; we have to determine

SECT. I.] TWO PROBLEMS. 335

what is the state of the line after a given time. This problem
may be made more general, by supposing, 1st, that the initial
temperatures of the points included between a and b are unequal
and represented by the ordinates of any line whatever, which we
shall regard first as composed of two symmetrical parts (see fig. 16);

Fig. 16.

2nd, that part of the heat is dispersed through the surface of the
solid, which is a prism of very small thickness, and of infinite
length.

-.JO*6 second problem consists in determining the successive
states of a prismatic bar, infinite in length, one extremity of
which is submitted to a constant temperature. The solution of
these two problems depends on the integration of the equation

dv _ K tfv HL
dt~CDdxz CDSV'

(Article 105), which expresses the linear movement of heat, v is
the temperature which the point at distance x from the origin
must have after the lapse of the time t ; K, H, C, D, L, S, denote
the internal and surface conducibilities, the specific capacity for
heat, the density, the contour of the perpendicular section, and
the area of this section.

345. Consider in the first instance the case in which heat is
propagated freely in an infinite line, one part of which ab has
received any initial temperatures; all other points having the
initial temperature 0. If at each point of the bar we raise the
ordinate of a plane curve so as to represent the actual tempera
ture at that point, we see that after a certain value of the time t,
the state of the solid is expressed by the form of the curve.
Denote by v = F(x) the equation which corresponds to the given
initial state, and first, for the sake of making the investigation

336 THEORY OF HEAT. [CHAP. IX.

more simple, suppose the initial form of the curve to be composed
of two symmetrical parts, so that we have the condition

F(x)=F(-x}.

JL-i. HL -

CD~ ' CDS~
dt^

in the equation ~rr — k-j—2 — hv, make v = e~htu, and we have

du -, d*u

dt dz* ' Jcv

\

Assume a particular value of u, namely, a cos qx e"^1 ; a and q
being arbitrary constants. Let qv q2, q3, &c. be a series of any
values whatever, and al9 a2, a3, &c. a series of corresponding
values of the coefficient Q, we have

u = al cos fax) e~*<zi2< + a2 cos faai) e~kq^ + aa cos fax) e-^* + &c.
Suppose first that the values qlt q^, qs, &c. increase by infinitely
small degrees, as the abscissa q of a certain curve ; so that they
become equal to dq, 2dq, 3dq> &c. ; dq being the constant differen
tial of the abscissa; next that the values a^ a2, a3> &c. are pro
portional to the ordinates Q of the same curve, and that they
become equal to Q^dq, Q^dq, Q3dq, &c., Q being a certain function
of q. It follows from this that the value of u may be expressed
thus :

u = Idq Q cos qx e~ktjH}

Q is an arbitrary function f(q), and the integral may be taken
from q — Q to q=vo. The difficulty is reduced to determining
suitably the function Q.

346. To determine Q, we must suppose t — 0 in the expression
for u, and equate u to F (x). We have therefore the equation of
condition

If we substituted for Q any function of q, and conducted the
integration from q = 0 to q = oo, we should find a function of x :

it is required to solve the inverse problem, that is to say, to
ascertain whatranctioii of q, after being substituted for Q, gives
as the result the function F(x)t a remarkable problem whose
solution demands attentive examination.

SECT. I.] AN INVERSE PROBLEM. 337

Developing the sign of the integral, we write as follows, the
equation from which the value of Q must be derived :

F(x) = dq Qt cos qjc + dqQz cos qzx + dqQ3 cos qzx + &c.

In order to make all the terms of the second member dis
appear, except one, multiply each side by dxcosrx, and then
integrate with respect to x from x = 0 to x — mr, where n is an
infinite number, and r represents a magnitude equal to any one
of qlf qz, q3, &c., or which is the same thing dq, 2dq, 3dq, &c. Let
qi be any value whatever of the variable qf and q^ another value,
namely, that which we have taken for r; we shall have r =jdq,
and q = idq. Consider then the infinite number n to express how
many times unit of length contains the element dq, so that we

have n = -r- . Proceeding to the integration we find that the
dq

value of the integral Idx cos qx cos rx is nothing, whenever r and

q have different magnitudes ; but its value is ^ UTT, when q = r.

This follows from the fact that integration eliminates from the
second member all the terms, except one ; namely, that which
contains qj or r. The function which affects the same term
is Qj, we have therefore

dx F (x) cos qx = dq Q} ^ nir,
and substituting for ndq its value 1, we have

cos qx.

Q (•*>

We find then, in general, -^ = dxF(x)cosqx. Thus, to

2 Jo

determine the function Q which satisfies the proposed condition,
we must multiply the given function F(x) by dxcosqx, and in-

2
tegrate from x nothing to x infinite, multiplying the result by - ;

that is to say, from the equation F(x] = ldqf(q) cos qx, we deduce

2 r

f(q}=-ld,jcF(x)cosqx, the function F(f) representing the
F. ii. 22

338 THEORY OF HEAT. [CHAP. IX.

initial temperatures of an infinite prism, of which an intermediate
part only is heated. Substituting the value of/(^) in the expres
sion for F (x}y we obtain the general equation

— F(x)=\ dqcosqxl dxF(x)cv$qx (e).

A Jo Jo

347. If we substitute in the expression for v the value which
we have found for the function Q, we have the following integral,
which contains the complete solution of the proposed problem,

-v • ^a

7I— = e~u \ dq cos qx e~kqH I dx F (x) cos qx.

° °.

The integral, with respect to #, being taken from x nothing

fcy* to x infinite, the result is a function of q\ and taking then the
integral with respect to q from q = 0 to q = oo , we obtain for v a
function of x and t, which represents the successive states of the
solid. Since the integration with respect to x makes this variable
disappear, it may be replaced in the expression of v by any varia
ble a, the integral being taken between the same limits, namely
from a = 0 to a = oo . We have then

!L_ _. e-u I fa cos gX e-kq*t I fa 2P(fl). cos qx,
Jo Jo

or — = e~ht I dx F(a.) I dq e~kqZf cos qx cos qy.

a Jo Jo

\

The integration with respect to q will give a function of x}

t and a, and taking the integral with respect to a we find a func-
^ tion of x and t only. In the last equation it would be easy to
effect the integration with respect to q, and thus the expression
of v would be changed. We can in general give different forms
to the integral of the equation

dv , d*v ,

dt=kdJ?~hv' •<•$•"

they all represent the same function of x and t.

348. Suppose in the first place that all the initial tempera
tures of points included between a and b, from x = — 1, to x — 1,
have the common value 1, and that the temperatures of all the

SECT. I.] FUNCTIONS EXPRESSED BY INTEGRALS. 339

other points are nothing, the function F(x) will be given by this
condition. It will then be necessary to integrate, with respect to
x, from x — 0 to x = 1, for the rest of the integral is nothing
according to the hypothesis. We shall thus find

~ 2 sin q , irv , . C^dg , 2,

0 = ---- * and -TT = e~M I — e q cos qx sm a.

* 1 - JO 1 I

^ The second member may easily be converted into a convergent
series, as will be seen presently ; it represents exactly the state
of the solid at a given instant, and if we make in it t = 0, it ex
presses the initial state.

Thus the function — I — sin q cos qx is equivalent to unity, if \

we give to x any value included between — 1 and 1 : but this
function is nothing if to x any other value be given not included /
between — 1 and 1. We see by this that discontinuous functions / /
also may be expressed by definite integrals.

349. In order to give a second application of the preceding
formula, let us suppose the bar to have been heated at one of its
points by the constant action of the same source of heat, and
that it has arrived at its permanent state which is known to be
represented by a logarithmic curve.

It is required to ascertain according to what law the diffusion
of heat is effected after the source of heat is withdrawn. Denoting
by F ' (x) the initial value of the temperature, we shall have

/HL
F(x) = Ae A'^; A is the initial temperature of the point

most heated. To simplify the investigation let us make A = l,

TTT

and -^7=1. We have then F(x\—e~x, whence we deduce
Ao

Q = I dx e~x cos qx, and taking the integral from x nothing to x

innnite;;<§ =^j - 3. T
the following equation :

innnite;;<§ =^j - 3. Thus the value of v in x and t is given by

22—2

34-0 THEORY OF HEAT. [CHAP. IX.

350. If we make £=0, we have ~ = I •—. JM which cor-

£ Jo 1 + 2

responds to the initial state. Hence the expression - I — ^ -•£—

is equal to e-x. It must be remarked that the function F(x),
which represents the initial state, does not change its value accord
ing to hypothesis when x becomes negative. The heat communi
cated by the source before the initial state was formed, is
propagated equally to the right and the left of the point 0, which
directly receives it: it follows that the line whose equation is

2 f^dqcoaqx . , f . • i i ^ T-I

y = — I — = 2" 1S composed ot two symmetrical branches whicii

are formed by repeating to right and left of the axis of y the part
of the logarithmic curve which is on the right of the axis of y, and
whose equation is y = e~x. We see here a second example of a
discontinuous function expressed by a definite integral. This

function - I ^ C°S f^- is equivalent to e~x when x is positive, but
it is ex when x is negative1.

351. The problem of the propagation of heat in an infinite
bar, one end of which is subject to a constant temperature, is
reducible, as we shall see presently, to that of the diffusion of heat
in an infinite line; but it must be supposed that the initial heat,
instead of affecting equally the two contiguous halves of the solid,
is distributed in it in contrary manner; that is to say that repre
senting by F(x) the temperature of a point whose distance from
the middle of the line is x, the initial temperature of the opposite
point for which the distance is — &, has for value — F (x).

This second problem differs very little from the preceding, and
might be solved by a similar method: but the solution may
also be derived from the analysis which has served to determine
for us the movement of heat in solids of finite dimensions.

Suppose that a part ab of the infinite prismatic bar has been
heated in any manner, see fig. (16*), and that the opposite part
a/3 is in like state, but of contrary sign ; all the rest of the solid
having the initial temperature 0. Suppose also that the surround-

1 Of. Biemann, Part. Diff. Glcich. § 16, p. 34. [A. F.]

SECT. I.] HEATED FINITE BAR. 841

ing medium is maintained at the constant temperature 0, and that
it receives heat from the bar or communicates heat to it through

Fig. 16*.

the external surface. It is required to find, after a given time t>
what will be the temperature v of a point whose distance from the
origin is x.

We shall consider first the heated bar as having a finite
length 2JT, and as being submitted to some external cause which
maintains its two ends at the constant temperature 0; we shall
then make JT= oc.

352. We first employ the equation

' r

and makin v = e~hfu we have

_ ,
dt ~ dx*>

the general value of u may be expressed as follows :

u = aie~ k9iH sin gjc + agr*^ sin gjc + a&e ~ *0& sin gax -f &c.

Making then x = X, which ought to make the value of v
nothing, we have, to determine the series of exponents g, the
condition sin gX= 0, or gX=i7r, i being an integer.

Hence

• • . ^

u * •=. a^e sin -^ + a2e sin — =- + &c.

It remains only to find the series of constants alt aa, a3, &c.
Making t = 0 we have

. . . „

sin -.+ a sin -- + a3 sin -«- + «xc.

342 THEORY OF HEAT. [CHAP. IX.

Let ~Y — r, and denote F (x) or F(—- j by f(r) ; we have
f(r) = «j sin r + «2 sin 2r + aa sin 3r -f &c.

2 r

Now, we have previously found a» = - \drf(r) sinir, the inte
gral being taken from r = 0 to r = TT. Hence

The integral with respect to x must be taken from x = 0 to
x = X Making these substitutions, we form the equation

sin

353. Such would be the solution if the prism had a finite
length represented by 2X. It is an evident consequence of the
principles which we have laid down up to this point; it remains
only to suppose the dimension X infinite. Let X= UTT, n being
an infinite number; also let q be a variable whose infinitely small

increments dgr are all equal ; we write -7- instead of n. The general

term of the series which enters into equation (a) being

. ITTX , —..

sin -- ,

jpi2* . ITTX ( , —..
sm^jdxF (x)

we represent by 3- the number i, which is variable and becomes
infinite. Thus we have

-v IT 1 . q

JL = -T-, n = -7- , fc=-j-.
dy dq dqr

Making these substitutions in the term in question we find
e~kqH sin gx\dxF (x) sin qx. Each of these terms must be divided

*7T

by X or v-, becoming thereby an infinitely small quantity, and

SECT. I.] GENERAL SOLUTION. 343

the sum of the series is simply an integral, which must be taken
with respect to q from q = 0 to q = oo . Hence

v — - e~M \dqe-W* sin qx \dxF(x)smqx ......... (a),

the integral with respect to x must be taken from x = 0 to x = oo.
We may also write

TTl) f30 f°

_ Q-U \ dqe-Wt sm qx I
* Jo Jo

7TV f30 f30

~^—Q~u\ d^F(^]\ dq e-
* Jo Jo

sm

or

Equation (a) contains the general solution of the problem;
and, substituting for F(x] any function whatever, subject or not
to a continuous law, we shall always be able to express the value
of the temperature in terms of x and t : only it must be remarked
that the function F(x) corresponds to a line formed of two equal
and alternate parts1.

354. If the initial heat is distributed in the prism in such a
manner that the line FFFF (fig. 17), which represents the initial

Fig. 17.

state, is formed of two equal ares situated right and left of
the fixed point 0, the variable movement of the heat is expressed
by the equation

TTV f30 f00

-_ = e~u I -d&F(a) I dq e~W cos qx cos ga.

Fig. 18.

If the line ffff (fig. 18), which represents the initial state, is
i That is to say, F(x)=-F(-x}. [A.F.]

344 THEORY OF HEAT. [df AP. IX.

formed of two similar and alternate arcs, the integral which gives
the value of the temperature is

TTV
¥

Too Too

= e~u \ dxf(a) da e~kqHs'm qx sin qa..
Jo Jo

If we suppose the initial heat to be distributed in any manner,
it will be easy to derive the expression for v from the two preced
ing solutions. In fact, whatever the function $ (x) may be, which
represents the given initial temperature, it can always be decom
posed into two others F (x) +/(#), one of which corresponds to the
line FFFF, and the other to the \iueffff, so that we have these
three conditions

F(x) = *•(-*),/(*) = -/(- *), <}> («) = F(x) +f(x).

We have already made use of this remark in Articles 233 and
234. We know also that each initial state gives rise to a variable
partial state which is formed as if it alone existed. The composi
tion of these different states introduces no change into the tem
peratures which would have occurred separately from each of
them. It follows from this that denoting by v the variable tem
perature produced by the initial state which represents the total
function cf> (x), we must have

-. / r°° r°°

_ e-u M fa g-*a^COs qx I dot. F (a) cos qy.

4 WO Jo

+ 1 dq e-**** sin. qx I da/(a) sin qaj.
Jo Jo '

If we took the integrals with respect to a between the limits
— oo and + oo , it is evident that we should double the results.
We may then, in the preceding equation, omit from the first
member the denominator 2, and take the integrals with respect to
a in the second form a = — oo toa = + oo. We easily see also

r+<x> r+oo

that we could write I da $ (a) cos ga, instead of I da. F(a) cos qy. ;

J —00 J - 00

for it follows from the condition to which the function /(a) is sub
ject, that we must have

r+ao
0 = I daf(ot) cosqy.

J -oo

SECT. I.] ANY INITIAL DISTRIBUTION. 345

We can also write

f+ao r+oo ? f**^

\ dj. <f> (a) sin qy. instead of I dif(o.} ee*s qx,

J -oo J -oo

for we evidently have

0= [ "diFtynnqx,

J — oo

We conclude from this

Too / r+oo

TTV = e~ht\ dq Q-W-t I da. $ (a) cos qy. cos qx

JO V J - oo

+ 1 da<£ (ajsin^sinja;) ,

J -00 /

/•oo /•+«

or, 7rv = e-Ml dqe~k<*H dx 0 (a) cos ^ (« — a),

JO J -oo

r + oo Too

or, 7rv=e~ht\ dz<l>(oL) I dqe-k*2tcosq (x — a).

J -oo Jo

355. The solution of this second problem indicates clearly
what the relation is between the definite integrals which we have
just employed, and the results of the analysis which we have
applied to solids of a definite form. When, in the convergent
series which this analysis furnishes, we give to the quantities
which denote the dimensions infinite values ; each of the
terms becomes infinitely small, and the sum of the series is
nothing but an integral. We might pass directly in the same
manner and without any physical considerations from the different
trigonometrical series which we have employed in Chapter ill. to
definite integrals ; it will be sufficient to give some examples of
these transformations in which the results are remarkable.

356. In the equation

7 TT = sin u + ^ sin 3z* + ~ sin ou + &c.
4 3 o

/yi

we shall write instead of u the quantity - ; x is a new variable,

and n is an infinite number equal to -=- ; q is a quantity formed by
the successive addition of infinitely small parts equal to dq. We

34G THEORY OF HEAT. [CHAP. IX.

shall represent the variable number i by -J- . If in the general

term —. sin (2* + 1) -we put for i and n their values, the term

2^ + 1 ' n

becomes ^sin2<7#. Hence the sum of the series is \ ~sm2qx,
2q • J $

the integral being taken from q = 0 to q = oo ; we have therefore

the equation \ IT = J I — sin 2qx which is always true whatever

Jo %

be the positive value of x. Let 2qx = r, r being a new varia
ble, we have — = — and J TT = I - sin r ; this value of the defi
nite integral I — sin r has been known for some time. If on

supposing r negative we took the same integral from r = 0 to
r = — oo , we should evidently have a result of contrary sign — -J TT.

357. The remark which we have just made on the value of
the integral I — sin r, which is J TT or — \ TT, serves to make known
the nature of the expression

2 f^dqsi]
*h~^l

cos qxy

whose value we have already found (Article 348) to be equal to
1 or 0 according as x is or is not included between 1 and — 1.

"We have in fact

I — cos qx sin q = J I — sin ^ (x 4- 1) — £ I — sin q (x — 1) ;
the first term is equal to J TT or — J TT according as x + 1 is a
positive or negative quantity; the second J I — sin q (x — 1) is equal

to J TT or — J TT, according as x — 1 is a positive or negative quantity.
Hence the whole integral is nothing if x + 1 and x — 1 have the
same sign ; for, in this case, the two terms cancel each other. But
if these quantities are of different sign, that is to say if we have at
the same time

x -f 1 > 0 and x — 1 < 0,

SECT. I.]

PROPERTIES OF DEFINITE INTEGRALS.

347

the two terms add together and the value of the integral is J TT.

Hence the definite integral1 - — sin a cos qx is a function of x

vrJo q

equal to 1 if the variable x has any value included between 1 and
— 1 ; and the same function is nothing for every other value of x
not included between the limits 1 and — 1.

358. We might deduce also from the transformation of series
into integrals the properties of the two expressions2

2 r°° dq cos qx , 2 f °° qdq sin qx t
vJt 1 + <f ' FC W0 1 + 22 '

the first (Art. 350) is equivalent to e~x when x is positive, and to
ex when x is negative. The second is equivalent to e~x if x is positive,
and to — ex if x is negative, so that the two integrals have the
same value, when x is positive, and have values of contrary sign
when x is negative. One is represented by the line eeee (fig. 19),
the other by the line eeee (fig. 20).

Fig. 19. Fig. 20.

The equation
1 . TTX __ sin a sin x sin 2a sin 2# sin 3 a sin 3x

£> olLL ™^ o v "T" 2 V' 2 •" 2 O52 2 ~1™ O^Cij

which we have arrived at (Art. 226), gives immediately the integral

2 f™ dqsinqTTsmqx ,., 3. . , . ..

- I — — ^ — 2 — ? which expression is equivalent to sin x, if x

is included between 0 and TT, and its value is 0 whenever x ex
ceeds 7T.

1 At the limiting values of x the value of this integral is | ; Eiemann, § 15.

2 Cf. Eiemann, § 16.

3 The substitutions required in the equation are — for », dq for -, q for »'-.

We then have sin x equal to a series equivalent to the above integral for values of x
between 0 and TT, the original equation being true for values of x between 0 and a.

[A.F.]

348 THEORY OF HEAT. [CHAP. IX.

359. The same transformation applies to the general equation
£ TT cf> (w) = sin u Idu <f>(u)smu+ sin 2w Idu $ (u) sin 2w + &c.

/y» /'/*\

Making w = - , denote $ (w) or $ (-) by /(a?), and introduce into

ft \%/

the analysis a quantity ^ which receives infinitely small incre
ments equal to dq, n will be equal to -j- and i to ~ ; substituting
these values in the general term

. ix [dx . fx\ . ix
sin — I — d> ( - sin — ,
n J n r \nj n

we find dq smqx I dxf (x} sin qx. The integral with respect to u

is taken from u = 0 to u = TT, hence the integration with respect to
x must be taken from x = 0 to x = n?r, or from x nothing to x
infinite.

We thus obtain a general result expressed by the equation

Too Too

J •/(*)"* I djnn«pj dxf(x)smqx (e),

^o ^o

for which reason, denoting by Q a function of q such that we have
f(u)=ldqQsmqu an equation in which /(it) is a given function,

2 f
we shall have Q = - lduf(u) sinqu, the integral being taken from

u nothing to u infinite. We have already solved a similar problem
(Art. 346) and proved the general equation

Too /•<*>

^irF(x} — \ dqcosqxl dxF(x)cosqx (e),

*o ' 'Jo

which is analogous to the preceding.

360. To give an application of these theorems, let us suppose
f(x)=xr, the second member of equation (e) by this substitution

becomes Idq sin qx Idx sin qx of.
The integral

jdx sin qx x* or —^ Iqdx sin qx (qx}r

SECT. I.] CERTAIN DEFINITE INTEGRALS. 349

is equivalent to -^ldusmuur, the integral being taken from u
nothing to u infinite.
Let fjL be the integral

00

du sin u ur ;
o

it remains to form the integral

L ' * rfj

I a q sin qx -^ LL, or LLX \ du sin u i

J q J

denoting the last integral by v, taken from u nothing to u infinite,
we have as the result of two successive integrations the term
xr fjiv. We must then have, according to the condition expressed
by the equation (e),

•| 7T Of = fJLV Xf Or JJLV — £ 7T J

thus the product of the two transcendants

/*°°, r . , [xdu ._ . . .

I aww smw and I — u sm w is ^TT.
Jo Jo u

For example, if r = - ^ , we find the known result

in the same manner we find

[™ducosu

I -7^-= 2

Jo ^/u V 2

and from these two equations we might also conclude the following1,

f°° 1 -

I dqe~q = g -S/TT, which has been employed for some time.

361. By means of the equations (e) and (e) we may solve the
following problem, which belongs also to partial differential
analysis. What function Q of the variable q must be placed under

1 The way is simply to use the expressions e~'= +cos ^-12+ */ -1 sin^/- 1 2,

transforming a and 6 by writing y* for t< and recollecting that \ - —

Cf. § 407. [R. I . E.]

350 THEORY OF HEAT. [CHAP. IX.

the integral sign in order that the expression I dqQe~qx may be

equal to a given function, the integral being taken from q nothing
to q infinite1? But without stopping for different consequences,
the examination of which would remove us from our chief object,
we shall limit ourselves to the following result, which is obtained
by combining the two equations (e) and (e).

They may be put under the form

- 7rf(x) = I dq sin qx I dzf (a) sin qx,

A * Jo Jo

1 /-co roo

and ~ TrF (x) = I dq cos qx daF (a) cos qx.

* Jo' "Jo

If we took the integrals with respect to a. from — oo to -f oo,
the result of each integration would be doubled, which is a neces
sary consequence of the two conditions

/(«) = -/(-«) and F(*)=F (-a).
We have therefore the two equations

-CO ,00

7rf(x) = I dq sin qx I dxf(<x) sin qx,
Jo J-»

,00 -00

and TrF (x) = I dq cos qx I r/aF(a) cos qx.

JO J-oo

We have remarked previously that any function $ (x) can
always be decomposed into two others, one of which F (x) satisfies
the condition F(x) — F(—x], and the other f(x) satisfies the
condition /(#) =— /(— x). We have thus the two equations

/+oo /H-oo

dzF (a) sin ^a, and 0 = I dxf(oL) cos qx,
-oo J -oo

1 To do this write ±x*J - 1 in f(x) and add, therefore

2 JQ, cos qx dq =f (x J~l) +f(-x A^l),
which remains the same on writing - x for x,

therefore Q = - jdx [f(x,J~l} +f(-x J^l)] cos qx dx.

Again we may subtract and use the sine hut the difficulty of dealing with
imaginary quantities recurs continually. [R. L. E.]

SECT, i.] FOURIER'S THEOREM. 351

whence we conclude

/-» -+00

TT [F(x) +/(#)] = TT<J> (x) = dq sin qx \ cZa/(a) sin qy.

JO" J -oo

/.» /•+<»

4- I dq cos £# I dzF (a) cos ^or,

JO J - oo

and TT<£ (a?) = I d^ gin %% I dx(j> (a) sin qa.

JO J-oo

.00 -+W

+ dg cos <?# I dz<j) (a) cos £ x,
Jo ' /'-*

or w$(#) = | di<f>(a)l dq(8mqx6

J - 00 / t "*

or lastly1, f (*) ->~ f d*4> W f c!qcosq(x-a)

TTj-oo JO "

The integration with respect to q gives a function of x and
a, and the second integration makes the variable a disappear.

Thus the function represented by the definite integral Idqcosq (x— a)

has the singular property, that if we multiply it by any function
</> (a) and by dx, and integrate it with respect to a between infinite
limits, the result is equal to TTCJ) (x) ; so that the effect of the inte
gration is to change a into a?, and to multiply by the number IT.

362. We might deduce equation (E) directly from the theorem

1 Poisson, in his Memoire sur la Theorie des Ondes, in iheMemoires de V Academic
dcs Sciences, Tome i. , Paris, 1818, pp. 85 — 87, first gave a direct proof of the theorem

1 00 -(-so

f(x) = - r dq r da e~k^ cos (gx - qa)f(a),

in which k is supposed to be a small positive quantity which is made equal to 0
after the integrations.

Boole, On the Analysis of Discontinuous Functions, in the Transactions of the
fioyal Irish Academy, Vol. xxi., Dublin, 1848, pp. 126—130, introduces some ana
lytical representations of discontinuity, and regards Fourier's Theorem as unproved
unless equivalent to the above proposition.

Deners, at the end of a Note sur quelques integrates definies &c., in the Bulletin
des Sciences, Societe Philomatique, Paris, 1819, pp. 161 — 166, indicates a proof of
Fourier's Theorem, which Poisson repeats in a modified form in the Journal Pobj-
technique, Cahier 19, p. 454. The special difficulties of this proof have been
noticed by De Morgan, Differential and Integral Calculus, pp. 619, 628.

An excellent discussion of the class of proofs here alluded to is given by
Mr J. W. L. Glaisher in an article On sinac and cos oo , Messenger of Mathematics,
Ser. i., Vol. v., pp. 232—244, Cambridge, 1871. [A. F.]

352 THEORY OF HEAT. [CHAP. IX.

stated in Article 2:34, which gives the development of any func
tion F(x) in a series of sines and cosines of multiple arcs. We
pass from the last proposition to those which we have just demon
strated, by giving an infinite value to the dimensions. Each term
of the series becomes in this case a differential quantity1. Trans
formations of functions into trigonometrical series are some of the
elements of the analytical theory of heat; it is indispensable to
make use of them to solve the problems which depend on this
theory.

The reduction of arbitrary functions into definite integrals,
such as are expressed by equation (E), and the two elementary
equations from which it is derived, give rise to different conse
quences which are omitted here since they have a less direct rela
tion with the physical problem. We shall only remark that the
same equations present themselves sometimes in analysis under
other forms. We obtain for example this result

1 r°° r°°
<j>(x)=- drf (a) I dqcosq(x — a) (Ef)

TfJ 0 JO

which differs from equation (E) in that the limits taken with
respect to a are 0 and oo instead of being — oo and + oo .

In this case it must be remarked that the two equations (E)
and (E') give equal values for the second member when the
variable x is positive. If this variable is negative, equation (E1)
always gives a nul value for the second member. The same is
not the case with equation (E), whose second member is equiva
lent to 7T(j) (x), whether we give to x a positive or negative value.
As to equation (E') it solves the following problem. To find a
function of x such that if x is positive, the value of the function
may be </> (x), and if x is negative the value of the function may
be always nothing2.

363. The problem of the propagation of heat in an infinite
line may besides be solved by giving to the integral of the partial
differential equation a different form which we shall indicate in

1 Eiemann, Part. Diff. Gleich. § 32, gives the proof, and deduces the formulae
corresponding to the cases F (x) = ± F ( - x).

2 These remarks are essential to clearness of view. The equations from which
(E) and its cognate form may be derived will be found in Todhunter's Integral
Calculus, Cambridge, 1862, § 316, Equations (3) and (4). [A. F.]

SECT. I.] VARYING TEMPERATURE IX IX FINITE BAR. 333

the following article. We shall first examine the case in which
the source of heat is constant.

Suppose that, the initial heat being distributed in any manner
throughout the infinite bar, we maintain the section A at a
constant temperature, whilst part of the heat communicated is dis
persed through the external surface. It is required to determine
the state of the prism after a given time, which is the object of the
second problem that we have proposed to ourselves. Denoting by
1 the constant temperature of the end A, by 0 that of the medium,

W^
we have e £S as the expression of the final temperature of a

point situated at the distance x from this extremity, or simply

TTJ-

e~xj assuming for simplicity the quantity -y to be equal to unity.

Denoting by v the variable temperature of the same point after
the time t has elapsed, we have, to determine v, the equation

•dvct*v HL

_

let now v = e~ Ks+u,

du K d*a HL ,

vehftve

dit' , (TV

- = k

rr TT T

replacing by k and by h.

If we make u=e~htu we have -,- —Jc j-a: the value of u or

dt dxa

»W—

v — e Ks is that of the difference between the actual and the
final temperatures ; this difference u, which tends more and more
to vanish, and whose final value is nothing, is equivalent at first to

-W^

F(x)r-e *,

denoting by F (x) the initial temperature of a point situated at the

distance x. Let f(x) be the excess of the initial temperature over

F. H. '!:}

354 THEORY OF HEAT. [CHAP. IX.

the final temperature, we must find for u a function which satisfies

the equation -r^k-r^ — hu, and whose initial value is f(x), and
ctt cl/x

-x>JT^

final value 0. At the point A, or x = 0, the quantity v-e
has, by hypothesis, a constant value equal to 0. We see by this
that u represents an excess of heat which is at first accumulated in
the prism, and which then escapes, either by being propagated to
infinity, or by being scattered into the medium. Thus to represent
the effect which results from the uniform heating of the end A of
a line infinitely prolonged, we must imagine, 1st, that the line is
also prolonged to the left of the point A, and that each point
situated to the right is now affected with the initial excess of
temperature ; 2nd, that the other half of the line to the left of
the point A is in a contrary state ; so that a point situated at the
distance - x from the point A has the initial temperature — /(#) :
the heat then begins to move freely through the interior of the
bar, and to be scattered at the surface.

The point A preserves the temperature 0, and all the other
points arrive insensibly at the same state. In this manner we are
able to refer the case in which the external source incessantly com
municates new heat, to that in which the primitive heat is propa
gated through the interior of the solid. We might therefore solve
the proposed problem in the same manner as that of the diffusion
of heat, Articles 347 and 353; but in order to multiply methods of
solution in a matter thus new, we shall employ the integral under
a different form from that which we have considered up to this
point.

364. The equation -^ = k -7-3 is satisfied by supposing u equal

to e~x ekt. This function of x and t may also be put under the form
of a definite integral, which is very easily deduced from the known

value of ldqe~q\ We have in fact *j7r=]dqe~q*, when the integral
is taken from = -coto = +oo. We have therefore also

J JT— \dqe~

SECT. I.] SOLUTION OF THE LINEAR EQUATION. 355

b being any constant whatever and the limits of the integral the
same as before. From the equation

we conclude, by making 62 = kt

hence the preceding value of u or e~* ekt is equivalent to

we might also suppose u equal to the function

a and w being any two constants ; and we should find in the same
way that this function is equivalent to

We can therefore in general take as the value of u the sum of an
infinite number of such values, and we shall have

+ &c.)

The constants alt a2, a3, &c., and nv nz, ns> &c. being undetermined,
the series represents any function whatever of x 4- Zg_>Jkt ; we have

therefore u= ldqe~qi^> (x + fyjkfy The integral should be taken
from 2r = — cotog'ss+x, and the value of u will necessarily satisfy
the equation -j- = k -y-j . This integral which contains one arbi

trary function was not known when we had undertaken our re
searches on the theory of heat, which were transmitted to the
Institute of France in the month of December, 1807: it has been

23—2

356 THEORY OF HEAT. [CHAP. IX.

given by M. Laplace1, in a work which forms part of volume vui
of the Me'moires de 1'Ecole Polytechnique ; we apply it simply to
the determination of the linear movement of heat. From it we
conclude

,, f+0°, 2JL/

y — g-hti dqe-V([>(x +

J -00

when t = 0 the value of u is F(x) — e
hence

= r

J _

and <> x = =

Thus the arbitrary function which enters into the integral, is deter
mined by means of the given function /(a?), and we have the
following equation, which contains the solution of the problem,

/WL e~M f+0° ,—

v = -^e *» + -7=- dqe-^f (x + Sta/ftj) , .

V 7T •/ _oo

it is easy to represent this solution by a construction.

365. Let us apply the previous solution to the case in which
all points of the line AB having the initial temperature 0, the end
A. is heated so as to be maintained continually at the tempera
ture 1. It follows from this that F (x) has a nul value when x

-x !^~L
differs from 0. Thus f(x} is equal to — e KS whenever x differs

from 0, and to 0 when x is nothing. On the other hand it is
necessary that on making x negative, the value off(x) should change
sign, so that we have the condition /(— x) ——f(x) • We thus
know the nature of the discontinuous function f(x)'t it becomes

. -

— e when x exceeds 0, and + e KS when x is less than 0.

We must now write instead of x the quantity x + 2q^kt. To find

r+co vi •

u orl dqe-* -.— f(x+ %VAtf),' we must first take the integral

from

= 0 to

1 Journal de TEcole Polytechnique, Tome vm. pp. 235—244, Paris, 1809.
Laplace shews also that the complete integral of the equation contains only one
arbitrary function, but in this respect he had been anticipated by Poisson. [A. F.J

SECT. I.] APPLICATION OF THE SOLUTION. 357

and then from

x + IqJkt = - oo to x + 2q*/ki = 0.
For the first part, we have

*>•

and replacing lc by its value -^ we have

VTT
-^•-</S/'

or _ Ji

TT T r

Denoting the quantity q + by r the preceding expression

becomes

e~Xlv^s ffu r

7=— ecus \dre- r\

VTT J

this integral idre-^ must be taken by hypothesis from

* + 22y "^==0

to

= 00,

/yi

or from a = — - 7= to a = oo ,

9

a;

,
or from r = - __ to r =

iKi

VCD

The second part of the integral is

«,

358

THEORY OF HEAT.

[CHAP. IX.

or

or

VTT

denoting by r the quantity q — A/
must be taken by hypothesis from

or from

from

-oo to

2

I jf+

> / XI- 6

VCD

The two last limits may, from the nature of the function e~r<i, be
replaced by these:

~HU . x

rKt
CD

r

• , and r = oo .

It follows from this that the value of u is expressed thus :

/ffi PU r Q IHL iTLt r

u = e * KS eCDS dre" — e~w ^ eona idre-^j
the first integral must be taken from

+ ,^- to r = oo ,

r =

and the second from

x

to r = co .

SECT. I.] FORM OF SOLUTION IN CASE CONSIDERED. 359

Let us represent now the integral — = Idre'^ from r = R to r = oo

JfJ
by T/T (R], and we shall have

HLt _, x

cDs+^jm

CD

y_ _HLt

hence u, which is equivalent to" eTcDS t is expressed by

and

The function denoted by i/r (7?) has been known for some time,
and we can easily calculate, either by means of convergent series,
or by continued fractions, the values which this function receives,
when we substitute for R given quantities; thus the numerical
application of the solution is subject to no difficulty1.

1 The following references are given by Riemann:

Kramp. Analyse des refractions astronomiques et terrestres. Leipsic and Paris,

An. vii. 4to. Table I. at the end contains the values of the integral / e'
from k = 0-00 to fc = 3 -00.

Legendre. Traite desfonctions elliptiques et des integrates Euleriennes. Tomen.

360 THEORY OF HEAT, [CHAP. IX.

366. If H be made nothing, we have

This equation represents the propagation of heat in an infinite
bar, all points of which were first at temperature 0, except those at
the extremity which is maintained at the constant temperature 1.
We suppose that heat cannot escape through the external surface
of the bar ; or, which is the same thing, that the thickness of the
bar is infinitely great. This value of v indicates therefore the law
according to which heat is propagated in a solid, terminated by
an infinite plane, supposing that this infinitely thick wall has first
at all parts a constant initial temperature 0, and that the surface is
submitted to a constant temperature 1. It will not be quite
useless to point out several results of this solution.

Denoting by 0 (7?) the integral —^ \dre~r* taken from r = 0 to

JTTJ

r = 7?, we have, when R is a positive quantity,

hence

(- 5) — ^> (JR) = 20 CR) and t? = l-20/ ~

~CD,
developing the integral 0 (R) we have

Paris, 1826. 4to. pp. 520—1. Table of the values of the integral Jdx (log IV*.

The first part for values of Hog - j from 0-00 to 0-50; the second part for values
of x from 0-80 to $-00.

Encke. Astronomisches Jahrbuchfvr 1834. Berlin, 1832, 8vo. Table I. at the

2 ft
end gives the values of - / e~tzdt from f = 0-00 to t = 2'QO. [A. F.]

SECT. I.] MOVEMENT ACROSS INFINITE PLANES. 361

hence

1st, if we suppose x nothing, we find v = 1 ; 2nd, if x not
"being nothing, we suppose t = 0, the sum of the terms which

contain x represents the integral \dre~** taken from r = 0 to r = oo ,

• -

and consequently is equal to \J-jr; therefore v is nothing; 3rd,

different points of the solid situated at different depths cclt xv #3,
&c. arrive at the same temperature after different times tlt tit t&,
&c. which are proportional to the squares of the lengths xlt a?2, xz,
&c.; 4th, in order to compare the quantities of heat which during
an infinitely small instant cross a section S situated in the interior
of the solid "at a distance x from the heated plane, we must take

the value of the quantity — KS r and we have

thus the expression of the quantity -T- is entirely disengaged from
the integral sign. The preceding value at the surface of the

/ /Hf/} T7"

heated solid becomes S _ - , which shews how the flow of heat

at the surface varies with the quantities C, D, K, t ; to find how
much heat the source communicates to the solid during the lapse
of the time t, we must take the integral

362 THEORY OF HEAT. [CHAP. IX.

=- or

thus the heat acquired increases proportionally to the square root of
the time elapsed.

367. By a similar analysis we may treat the problem of the
diffusion of heat, which also depends on the integration of the

equation ~r: = k j-^ — hv. Representing by f^x) the initial tem
perature of a point in the line situated at a distance x from the
origin, we proceed to determine what ought to be the temperature
of the same point after a time t. Making v = e~htz, we have

-y- = k -Tg- , and consequently z — I dq e~qt^> (x + 2q Jkt). When

(it Ut J -oo

t — 0, we must have

9 (x) or

J — GO

hence

e~ty

To apply this general expression to the case in which a part of
the line from x — — ato# = ais uniformly heated, all the rest of
the solid being at the temperature 0, we must consider that the
factor f(x+ 2q Jfo) which multiplies e~qZ has, according to hypo
thesis, a constant value 1, when the quantity which is under the
sign of the function is included between — a and a, and that all
the other values of this factor are nothing. Hence the integral

Idq e-v* ought to be taken from x+2q Jkt = — a to x + 2q JTt = a,

or from q= -—-j^.~ toq= . Denoting as above by -^ & (It)

**jkt *>Jkt VTT

the integral ldre~rZ taken from r = R to r = oo , we have

2jktn

SECT. I.] COOLING OF AN INFINITE BAR. 363

368. We shall next apply the general equation

7T J —

to the case in which the infinite bar, heated by a source of
constant intensity 1, has arrived at fixed temperatures and is
then cooling freely in a medium maintained at the temperature
0. For this purpose it is sufficient to remark that the initial

_XJ*
function denoted by f(x) is equivalent to e v* so long as the

variable x which is under the sign of the function is positive,

and that the same function is equivalent to e^* when the
variable which is affected by the symbol /is less than 0. Hence

the first integral must be taken from

x + 2q-Jkt = 0 to x + fy-Jkt = oo ,
and the second from

x + ZqjTtt - - oo to x + 207^ = 0.
The first part of the value of v is

e~ht fie r _

. Q-X\ jfc" {(JqQ — ^Q — ^fl^Jht

Jv J

or

or ^ .—"[dre-**',
making r = g' 4- ^/Ai. The integral should be taken from
2 = ^r to 2 = °° >

or from r = = to r

364 THEORY OF HEAT. [CHAP. IX.

The second part of the value of v is

n~-Tlt .- /• ,-- /•

-T^ex\f^ldq e-<? &&* or e V* dr e~* ;
making r = q — JTti. The integral should be taken from
r = — oo tor = — Jfa -- 7= ,

_ . /%•

or from r = Jht -f — j=. to r = co ,

>_ ^y/ Kit

whence we conclude the following expression :

3C9. We have obtained (Art. 367) the equation

to express the law of diffusion of heat in a bar of small thickness,
heated uniformly at its middle point between the given limits

x = — a, x — + a.

We had previously solved the same problem by following a
different method, and we had arrived, on supposing a = 1, at
the equation

_lcos qx sin ^e-«2^, (Art. 348).

To compare these two results we shall suppose in each x = 0 ;

denoting again by ^{R} the integral ldre~rZ taken from r = 0
to r = R, we have

_ 1 1 /o

: ~i 3

\" 1 1 / a y )

+ 5 l; -&ft ;

SECT. I.] IDENTITY OF DIFFERENT SOLUTIONS. 365

on the other hand we ought to have

v = ~ e~M I— sin q e~q*kf,
TT j

q

or v =

[8

Now the integral Icfo<e~w2w2m taken from u = Q to u = oo has

a known value, m being any positive integer. We have in
general

Jo

o -2222 2 2 V* •

The preceding equation gives then, on making q*kt = if,

T, [2/, u2 1 u* 1 \

\due~u 1 1 — 15 Ti + fr 7T3-- &C. I ,

J V 3/.-« o^ ;'

v ii/_j_y 1

+ ' :C' '

13 ,/fc [2 5 z

This equation is the same as the preceding when we suppose
a. = 1. We see by this that integrals which we have obtained
by different processes, lead to the same convergent series, and
we arrive thus at two identical results, whatever be the value
of x.

We might, in this problem as in the preceding, compare the
quantities of heat which, in a given instant, cross different
sections of the heated prism, and the general expression of these
quantities contains no sign of integration ; but passing by these
remarks, we shall terminate this section by the comparison of
the different forms which we have given to the integral of the
equation which represents the diffusion of heat in an infinite
line.

r>>-n m ,• c ., ,. dll ^ d*ll

3/0. lo satisfy the equation ~r — k ^ Z) we may assume

u = e~ff ekt, or in general u — e~n? en°'kt, whence we deduce easily
(Art. 364) the integral
r
u = I

1 Cf. Rieinann, § 18.

3G6 THEORY OF HEAT. [CHAP. IX.

From the known equation

we conclude

-+00

N/7r = / dqe~(q+a)\ a being any constant; we have therefore

-

i, or

This equation holds whatever be the value of a. We may de-
velope the first member; and by comparison of the terms we shall

obtain the already known values of the integral ldqe~q* qn. This

value is nothing when n is odd, and we find when n is an even
number 2w,

L

2.2.2.2...

371. We have employed previously as the integral of the

du , d?u ,,
equation -rr = k^ the expression

u — a^-nW cos n^x + aj3~n**kt cos n^x + aae~n**kt cos nBx + &c. ;
or this,

u — a^e"n^kt sin n^x -h a2e~n**kt sin nzx + a&e~n*lkt sin nax + &c.

a,, a2, as) &c,, and Wj, wa, nB, &c., being two series of arbitrary
constants. It is easy to see that each of these expressions is
equivalent to the integral

(dq e~q* sin n (x + 2q *Jkt), or Idq e~& cos n

In fact, to determine the value of the integral

r*30

dq e~^ sin

J — 06

SECT. I.] IDENTITY OF SOLUTIONS. 367

we shall give it the following form

Idq e~q* sin x cos 2q *Jkt + jdy e~^ cos x sin 2q ^ki ;
or else,

,P/ fe-M e-t

4- / da e~i cos x ft . _ — ^ f _ 1
./-„* V2V-1 2V- I/'

which is equivalent to

e-** sin x (±jdq e-(9- v-*02 + i /^ e-(</+ V-w>A

4- e-* cos a?

the integral ]dq *«**=** taken from ? = - x to ^ = x is V^

we have therefore for the value of the integral (dqe-«* sin (#+2? i/'kt),
the quantity VTT e~w sin a?,- and in general

VTT ern2*< sin w^ = J ^ e~^ sin n(x + 2q V^) ,
we could determine in the same manner the integral

,+«

I c?2 e-33 cos n (x + 2^ ^S) ,
the value of which is V? e'^1 cos ?i#.
We see by this that the integral
e-»W (a, sin n.a? + \ cos w.a?) + e~n*ki (a, sin w8a; + 62 cos w^)

+ e-""2^ (aa sin w3ic -f 63 cos up) -f &c.
is equivalent to

-i Cdq 9~* I*1 Sin Wl (iC + 22 V^) + aa sin w, (« 4- 2^ VS) + &c. |

v/7rj.» (^ cos Wl (a: + 2j V^) + 68 cos 7ia (x 4- 22 V^) 4- &cj §

368 THEORY OF HEAT. [CHAP. IX.

The value of the series represents, as we have seen previously,
any function whatever of x + 2q? *Jkt ; hence the general integral
can be expressed thus

= /

The integral of the equation -^- — &^— 2 may besides be pre
sented under diverse other forms1. All these expressions are
necessarily identical.

SECTION II.
Of the free movement of heat in an infinite solid.

372. The integral of the equation ,, = -^ -j-9 (a) furnishes

immediately that of the equation with four variables
dv

, , ,
.........

as we have already remarked in treating the question of the pro
pagation of heat in a solid cube. For which reason it is sufficient
in general to consider the effect of the diffusion in the linear
case. When the dimensions of bodies are not infinite, the distri
bution of heat is continually disturbed by the passage from the
solid medium to the elastic medium; or, to employ the expres
sions proper to analysis, the function which determines the
temperature must not only satisfy the partial differential equa
tion and the initial state, but is further subjected to conditions
which depend on the form of the surface. In this case the integral
has a form more difficult to ascertain, and we must examine the
problem with very much more care in order to pass from the case
of one linear co-ordinate to that of three orthogonal co-ordinates :
but when the solid mass is not interrupted, no accidental condition
opposes itself to the free diffusion of heat. Its movement is the
same in all directions.

1 See an article by Sir \V. Thomson, " On the Linear Motion of Heat," Part I,
Camb. Math. Journal, Vol. in. pp. 170—174. [A. F.]

SECT. IL] LINEAR MOVEMENT. 369

The variable temperature v of a point of an infinite line is
expressed by the equation

TT

a? denotes the distance between a fixed point 0, and the point m,
whose temperature is equal to v after the lapse of a time t. We
suppose that the heat cannot be dissipated through the external
surface of the infinite bar, and that the initial state of the bar is
expressed by the equation v=f(x). The differential equation,
which the value of v must satisfy, is

dt ~ CD dx*

But to simplify the investigation, we write
dv d*v

which assumes that we employ instead of t another unknown

i 4 Kt

equal to ^ .

If in/ (oj), a function of # and constants, we substitute X+%n*/t

for a:, and if, after having multiplied by -_ g-*2, we integrate with

VTT

respect to w between infinite limits, the expression

1 f+°°
^1 d?ie~na

satisfies, as we have proved above, the differential equation (b) ;
that is to say the expression has the property of giving the same
value for the second fluxion with respect to x} and for the first
fluxion with respect to t. From this it is evident that a function
of three variables f (x, y, z) will enjoy a like property, if we substi
tute for x, y, z the quantities

provided we integrate after having multiplied by

dn P-n* &L ,-*« *3Lf-q*

—j= e , ,- e * , ._ e * .

VTT VTT VTT

F. H. 24

370 THEORY OF HEAT. [CHAP. IX.

In fact, the function which we thus form,

gives three terms for the fluxion with respect to t, and these three
terms are those which would be found by taking the second fluxion
with respect to each of the three variables so, y, z.

Hence the equation

v = TT3 fdn jdpjdq

y +

gives a value of v which satisfies the partial differential equation
dv _ d*v d*v d*v .

~dt~dx^d^2 + ^"

373. Suppose now that a formless solid mass (that is to say
one which fills infinite space) contains a quantity of heat whose
actual distribution is known. Let v =F(x, y, z) be the equation
which expresses this initial and arbitrary state, so that the
molecule whose co-ordinates are x, y, z has an initial temperature
equal to the value of the given function F(x,y,z). We can
imagine that the initial heat is contained in a certain part of
the mass whose first state is given by means of the equation
v — F(xy y, z), and that all other points have a nul initial tem
perature.

It is required to ascertain what the system of temperatures
will be after a given time. The variable temperature v must
consequently be expressed by a function <j> (x, y, z, t) which ought
to satisfy the general equation (A) and the condition </> (x, y, z, 0)
= F(xt y, z}. Now the value of this function is given by the
integral

v = 7r

In fact, this function v satisfies the equation (A), and if in it we
make t = 0, we find

IT9 fdn j dp (dq e-W^+&F(x, y, z),
or, effecting the integrations, F (x, y, z).

SECT. II.] THE CASE OF THREE DIMENSIONS. 371

374. Since the function v or c/> (x, y, z, t] represents the
initial state when in it we make t = 0, and since it satisfies the
differential equation of the propagation of heat, it represents also
that state of the solid which exists at the commencement of the
second instant, and making the second state vary, we conclude
that the same function represents the third state of the solid, and
all the subsequent states. Thus the value of v, which we have
just determined, containing an entirely arbitrary function of three
variables x, y, z, gives the solution of the problem ; and we cannot
suppose that there is a more general expression, although other
wise the same integral may be put under very different forms.

Instead of employing the equation

we might give another form to the integral of the equation

-77 = -j-g ; and it would always be easy to deduce from it the
ctt dx

integral which belongs to the case of three dimensions. The
result which we should obtain would necessarily be the same as
the preceding.

To give an example of this investigation we shall make use of
the particular value which has aided us in forming the exponential
integral.

Taking then the equation -^- = ^-j ... (b), let us give to v the
very simple value e~nHcosnx, which evidently satisfies the
differential equation (6). In fact, we derive from it -j- = — rfv

d*v
and -y-g = — ri*v. Hence also, the integral

CUD

r

V — m

dn e~nZtcosnx

belongs to the equation (6) ; for this value of v is formed of the
sum of an infinity of particular values. Now, the integral

£

nx

24—2

372 THEORY OF HEAT. [CHAP. IX

f 3 Fri '

is known, and is known to be equivalent to /- /^ (see the follow
ing article). Hence this last function of x and t agrees also with
the differential equation (b). It is besides very easy to verify

_1J

P 4£

directly that the particular value -TF satisfies the equation in
question.

The same result will occur if we replace the variable x by
x — a, a being any constant. We may then employ as a particular

Q-q)2

value the function - &-j= — , in which we assign to a any value

whatever. Consequently the sum I dzf (a) - '—p — also satisfies

J v t>

the differential equation (6) ; for this sum is composed of an
infinity of particular values of the same form, multiplied by
arbitrary constants. Hence we can take as a value of v in the

//7) CM 7J

equation -j- = -3-- the following,
dt dx

A being a constant coefficient. If in the last integral we suppose
^ ' = j2, making also A — ~—r= , we shall have

1 f*00

V/^-oo

We see by this how the employment of the particular values

or

leads to the integral under a finite form.

SECT. II.] EVALUATION OF AN INTEGRAL. 373

375. The relation in which these two particular values are to
each other is discovered when we evaluate the integral1

/•

I

J

dn e'^t cos nx.

To effect the integration, we might develope the factor cos nx
and integrate with respect to n. We thus obtain a series which
represents a known development; but the result may be derived

more easily from the following analysis. The integral I dn e~n*£cos nx

is transformed to I dp e~^2cos 2pu, by assuming r?t =p2and nx = 2pu.
We thus have

/••foo 1 /•+«> -J.

I dn e~nH cos nx = —^l dp e~& cos 2pu. A

J -oo *JtJ -» /r^S

We shall now write ~S

Idpe~^cos2pu = ^ Idpe-^+fyu^-1 + \ f<#p e--pa-

~u* Idpe^-
--u* (dp e-

V

Now each of the integrals which enter into these two terms is
equal to A/TT. We have in fact in general

and consequently

= I

J -00

whatever be the constant b. We find then on making
b = T M s/^T, I ^ e"9' cos 2#w = e~tt' V^

hence I dn e~nH cos nx = - — ^ ,

j -oo *y^

1 The value is obtained by a different method in Todhunter's Integral Calcuhu,
§ 375. [A. F.]

374 THEORY OF HEAT. [CHAP. IX.

and putting for u its value —=> we have

2 V t

_£2

° e *t ,~

dn ff~*** cos nx = — — VTT.

pt

Moreover the particular value —j=- is simple enough to present

itself directly without its being necessary to deduce it from the
value e~nHcosnx. However it may be, it is certain that the

-& dv d*v

function —j=- satisfies the differential equation -j- = -^ • it is the

(j?~q)«

6~~ ^t

same consequently with the function — ^ — , whatever the quan-

*Jt

tity a may be.

376. To pass to the case of three dimensions, it is sufficient

_&M?

to multiply the function of x and t, — ^— , by two other similar

ijt

functions, one of y and t, the other of z and t\ the product will
evidently satisfy the equation

dv _ d*v d?v d?v
dt~d^ + dyz + d?'

We shall take then for v the value thus expressed :

If now we multiply the second member by den, d$, dy, and by
any function whatever/ (a, /3, 7) of the quantities a, /6, 7, we find,
on indicating the integration, a value of v formed of the sum of an
infinity of particular values multiplied by arbitrary constants.

It follows from this that the function v may be thus ex
pressed :

M-oo ,.+00 -+00 ^3 (q-^)2 + (.8-y)2+(Y-g)2

J-oo J-oo J -OP

This equation contains the general integral of the proposed
equation (A): the process which has led us to this integral oug^t^

SECT. II.] INTEGRAL FOR THREE DIMENSIONS. 375

to be remarked since it is applicable to a great variety of cases ; (
it is useful chiefly when the integral must satisfy conditions \
relative to the surface. If we examine it attentively we perceive I
that the transformations which it requires are all indicated by f
the physical nature of the problem. We can also, in equation (j)t
change the variables. By taking

we have, on multiplying the second member by a constant co
efficient A,

v = 23 A fdnfdp fdq erW+*'+f>f (x + 2n Jt, y + 2pji, z + 2$ Ji).

Taking the three integrals between the limits — oo and -f oo,
and making t = 0 in order to ascertain the initial state, we find

3

v = 23^7r~2/(#, y, z). Thus, if we represent the known initial
temperatures by F (x, y, z), and give to the constant A the value

-s _.§
2 TT 2, we arrive at the integral

8 r+x r+*> r+

v = 7r~2 dn\ dpi

J — oo J — oo J —

which is the same as that of Article 372.

The integral of equation (A) may be put under several other
forms, from which that is to be chosen which suits best the
problem which it is proposed to solve.

It must be observed in general, in these researches, that two
functions $ (as, y, z, t) are the same when they each satisfy the
differential equation (A), and when they are equal for a definite
value of the time. It follows from this principle that integrals,
which are reduced, when in them we make t = 0, to the same
arbitrary function F(x, y, z), all have the same degree of generality;
they are necessarily identical.

The second member of the differential equation (a) was

jr

multiplied by ^ , and in equation (6) we supposed this coefficient
equal to unity. To restore this quantity, it is sufficient to write

376 THEORY OF HEAT. [CHAP. IX.

Kt

TYT, instead of t, in the integral (i) or in the integral (f). We

\jJLJ

shall now indicate some of the results which follow from these
equations.

377. The function which serves as the exponent of the
number e* can only represent an absolute number, which follows
from the general principles of analysis, as we have proved ex
plicitly in Chapter II., section IX. If in this exponent we replace

Tfj.

the unknown t by 7^, we see that the dimensions of K} C, D and t,
(jU

with reference to unit of length, being — 1, 0, — 3, and 0, the

Kt
dimension of the denominator -^ is 2 the same as that of each

term of the numerator, so that the whole dimension of the expo
nent is 0. Let us consider the case in which the value of t increases
more and more; and to simplify this examination let us employ
first the equation

which represents the diffusion of heat in an infinite line. Suppose
the initial heat to be contained in a given portion of the line,
from x = — htox = +g, and that we assign to a? a definite value Xy
which fixes the position of a certain point m of that line. If the

time t increase without limit, the terms -r-r and — -— - which

4<t 4£

enter into the exponent will become smaller and smaller absolute

_*2 _2_o? _ft2

numbers, so that in the product e & e *t e & we can omit
the two last factors which sensibly coincide with unity. We thus
find

,, N
daf(a)

This is the expression of the variable state of the line after a
very long time ; it applies to all parts of the line which are less
distant from the origin than the point m. The definite integral

*2

* In such quantities as e~ * . [A. F.]

SECT. II.] INITIAL HEAT COLLECTED AT THE ORIGIN. 377

+ff

dnf(d) denotes the whole quantity of heat B contained in the

-h

solid, and we see that the primitive distribution has no influence
on the temperatures after a very long time. They depend only
on the sum B, and not on the law according to which the heat has
been distributed.

378. If we suppose a single element co situated at the origin
to have received the initial temperature/ and that all the others
had initially the temperature 0, the product cof will be equal to

r+ff
the integral I <fa/(a) or B. The constant /is exceedingly great

J —h

since we suppose the line co very small.

X*

The equation v = — ._ .. cof represents the movement which

2 »J TT *Jt

would take place, if a single element situated at the origin had
been heated. In fact, if we give to x any value a, not infinitely

X2

small, the function — - will be nothing when we suppose t = 0.

The same would not be the case if the value of x were

_-
nothing. In this case the function — — receives on the contrarv

an infinite value when t = 0. We can ascertain distinctly the
nature of this function, if we apply the general principles of the
theory of curved surfaces to the surface whose equation is

e~ty

g

The equation v = — ._ ._ a)f expresses then the variable tem-

perature at any point of the prism, when we suppose the whole
initial heat collected into a single element situated at the origin.
This hypothesis, although special, belongs to a general problem,
since after a sufficiently long time, the variable state of the solid is
always the same as if the initial heat had been collected at the
origin. The law according to which the heat was distributed, has

378 THEORY OF HEAT. [CHAP. IX.

much influence on the variable temperatures of the prism ; but
this effect becomes weaker and weaker, and ends with being quite
insensible.

379. It is necessary to remark that the reduced equation (i/)
does not apply to that part of the line which lies beyond the point
m whose distance has been denoted by X.

In fact, however great the value of the time may be, we might

2CLJ

choose a value of x such that the term e 4* would differ sensibly
from unity, so that this factor could not then be suppressed. We
must therefore imagine that we have marked on either side of the
origin 0 two points, m and m', situated at a certain distance X or
— X, and that we increase more and more the value of the time,
observing the successive states of the part of the line which is
included between m and m. These variable states converge more
and more towards that which is expressed by the equation

Whatever be the value assigned to X, we shall always be able to
find a value of the time so great that the state of the line mom
does not differ sensibly from that which the preceding equation (y)
expresses.

If we require that the same equation should apply to other
parts more distant from the origin, it will be necessary to suppose
a value of the time greater than the preceding.

The equation (?/) which expresses in all cases the final state of
any line, shews that after an exceedingly long time, the different
points acquire temperatures almost equal, and that the temperatures
of the same point end by varying in inverse ratio of the square
root of the times elapsed since the commencement of the diffusion.
The decrements of the temperature of any point whatever always
become proportional to the increments of the time.

380. If we made use of the interal

SECT. II.] ADMISSIBLE SIMPLIFICATIONS. 379

to ascertain the variable state of the points of the line situated at
a great distance from the heated portion, and in order to express

the ultimate condition suppressed also the factor e 4Jit , the
results which we should obtain would not be exact. In fact,
supposing that the heated portion extends only from a = 0 to a=g
and that the limit g is very small with respect to the distance x of
the point whose temperature we wish to determine ; the quantity

~~ 4kf whicn f°rms the exponent reduces in fact to — jy- ; that

(a _ xf xz
is to say the ratio of the two quantities , and ^- approaches

more nearly to unity as the value of x becomes greater with
respect to that of a : but it does not follow that we can replace
one of these quantities by the other in the exponent of e. In
general the omission of the subordinate terms cannot thus take
place in exponential or trigonometrical expressions. The quanti
ties arranged under the symbols of sine or cosine, or under the
exponential symbol ey are always absolute numbers, and we can
omit only the parts of those numbers whose value is extremely
small ; their relative values are here of no importance. To decide
if we may reduce the expression

rg (a-*)2 _^_ rg

&/(*)* ** toe H eZa/(a),

Jo J o

we must not examine whether the ratio of x to a is very great,
but whether the terms 77- > -TTI are very small numbers. This
condition always exists when t the time elapsed is extremely great ;

/y*

but it does not depend on the ratio - .

381. Suppose now that we wish to ascertain how much time
ought to elapse in order that the temperatures of the part of the
solid included between x — 0 and x = X, may be represented very
nearly by the reduced equation

380 THEORY OF HEAT. [CHAP. IX.

and that 0 and g may be the limits of the portion originally
heated.

The exact solution is given by the equation

(a-*)2

r^a/(a)e- «** ,.

1} = i , i A / ,

Jo Zirkt

and the approximate solution is given by the equation

(y),

k denoting the value ^j^ of the conducibility. In order that the
equation (y) may be substituted for the preceding equation (i}) it

2ax-a?

is in general requisite that the factor e *M , which is that which
we omit, should differ very little from unity ; for if it were 1 or \
we might apprehend an error equal to the value calculated or to

the half of that value. Let then e &* — 1 + w, to being a small

fraction, as ^^ or 77:7:7,; from this we derive the condition
LOO LOOO

a2\
I ,
J

— = a>, or t

co

and if the greatest value g which the variable a can receive is

1 O3C

very small with respect to x, we have t = - ^y .

co ^i/2

We see by this result that the more distant from the origin
the points are whose temperatures we wish to determine by means
of the reduced equation, the more necessary it is for the value of
the time elapsed to be great. Thus the heat tends more and more
to be distributed according to a law independent of the primitive
heating. After a certain time, the diffusion is sensibly effected,
that is to say the state of the solid depends on nothing more than
the quantity of the initial heat, and not on the distribution which
was made of it. The temperatures of points sufficiently near to
the origin are soon represented without error by the reduced
equation (y}\ but it is not the same with points very distant from

SECT. II.] NUMERICAL APPLICATION. 381

the source.* We can then make use of that equation only when
the time elapsed is extremely long. Numerical applications make
this remark more perceptible.

382. Suppose that the substance of which the prism is formed
is iron, and that the portion of the solid which has been heated is
a decimetre in length, so that g = O'l. If we wish to ascertain
what will be, after a given time, the temperature of a point m
whose distance from the origin is a metre, and if we employ for
this investigation the approximate integral (y), we shall commit
an error greater as the value of the time is smaller. This error
will be less than the hundredth part of the quantity sought, if the
time elapsed exceeds three days and a half.

In this case the distance included between the origin 0 and the
point in, whose temperature we are determining, is only ten times
greater than the portion heated. If this ratio is one hundred
instead of being ten, the reduced integral (y) will give the tem
perature nearly to less than one hundredth part, when the value
of the time elapsed exceeds one month. In order that the ap
proximation may be admissible, it is necessary in general, 1st that

2 2ft _ ft2

the quantity - -^ - should be equal to but a very small fraction

4/Lfc

as T~AA or TAAA or ^ess j 2nd, that the error which must follow
1UU

should have an absolute value very much less than the small
quantities which we observe with the most sensitive thermometers.
When the points which we consider are very distant from the
portion of the solid which was originally heated, the temperatures
which it is required to determine are extremely small ; thus the
error which we should commit in employing the reduced equation
would have a very small absolute value; but it does not follow
that we should be authorized to make use of that equation. For
if the error committed, although very small, exceeds or is equal to
the quantity sought ; or even if it is the half or the fourth, or an
appreciable part, the approximation ought to be rejected. It is
evident that in this case the approximate equation (y) would not
express the state of the solid, and that we could not avail ourselves
of it to determine the ratios of the simultaneous temperatures of
two or more points.

3~82 THEORY OF HEAT. [CHAP. IX.

383. It follows from this examination that we ought not to

1 W _(a-ff)»

conclude from the integral v = — 7= <fe/(a) e ~4*« " that the

law of the primitive distribution has no influence on the tempera
ture of points very distant from the origin. The resultant effect
of this distribution soon ceases to have influence on the points
near to the heated portion; that is to say their temperature
depends on nothing more than the quantity of the initial heat,
and not on the distribution which was made of it : but greatness
of distance does not concur to efface the impress of the distribu
tion, it preserves it on the contrary during a very long time
and retards the diffusion of heat. Thus the equation

only after an immense time represents the temperatures of points
extremely remote from the heated part. If we applied it without
this condition, we should find results double or triple of the true
results, or even incomparably greater or smaller; and this would
not only occur for very small values of the time, but for great
values, such as an hour, a day, a year. Lastly this expression
would be so much the less exact, all other things being equal, as
the points were more distant from the part originally heated.

384. When the diffusion of heat is effected in all directions,
the state of the solid is represented as we have seen by the
integral

If the initial heat is contained in a definite portion of the solid
mass, we know the limits which comprise this heated part, and
the quantities a, /3, 7, which vary under the integral sign, cannot
receive values which exceed those limits. Suppose then that we
mark on the three axes six points whose distances are + X, + Yf +Z,
and — X, — Y, — Z, and that we consider the successive states of
the solid included within the six planes which cross the axes at
these distances; we see that the exponent of e under the sign of

SECT. II.] APPROXIMATE FORMULA. 383

/g? J_ 7/2 _|_ 22\

integration, reduces to — f — ^- — — J, when tlie value of the time

increases without limit. In fact, the terms such as ^,- and ^r-

receive in this case very small absolute values, since the numera
tors are included between fixed limits, and the denominators
increase to infinity. Thus the factors which we omit differ
extremely little from unity. Hence the variable state of the
solid, after a great value of the time, is expressed by

The factor Idildft ldyf(z, /9, 7) represents the whole quantity

of heat B which the solid contains. Thus the system of tempera
tures depends .not upon the initial distribution of heat, but only
on its quantity. We might suppose that all the initial heat was
contained in a single prismatic element situated at the origin,
whose extremely small orthogonal dimensions were a)lt &>2, o>3. The
initial temperature of this element would be denoted by an
exceedingly great number /, and all the other molecules of the
solid would have a nul initial temperature. The product
G)ift)2Ct)3/ ig equal in this case to the integral

Whatever be the initial heating, the state of the solid which
corresponds to a very great value of the time, is the same as if all
the heat had been collected into a single element situated at the

385. Suppose now that we consider only the points of the
solid whose distance from the origin is very great with respect
to the dimensions of the heated part ; we might first imagine
that this condition is sufficient to reduce the exponent of e in
the general integral. The exponent is in fact

384 THEORY OF HEAT. [CHAP. IX.

and the variables a, /3, 7 are, by hypothesis, included between
finite limits, so that their values are always extremely small
with respect to the greater co-ordinate of a point very remote
from the origin. It follows from this that the exponent of e
is composed of two parts M+ p, one of which is very small
with respect to the other. But from the fact that the ratio

•^ is a very small fraction, we cannot conclude that the ex

ponential eH+* becomes equal to eM, or differs only from it by
a quantity very small with respect to its actual value. We must
not consider the relative values of M and JJL, but only the absolute
value of yLt. In order that we may be able to reduce the exact
integral (j) to the. equation

e m

' =jB

it is necessary that the quantity

2ao; + 2ffy + fyz - a* - ft2 - 7*

whose dimension is 0, should always be a very small number.
If we suppose that the distance from the origin to the point m,
whose temperature we wish to determine, is very great with
respect to the extent of the part which was at first heated,
we should examine whether the preceding quantity is always
a very small fraction «. This condition must be satisfied to
enable us to employ the approximate integral

but this equation does not represent the variable state of that
part of the mass which is very remote from the source of heat.
It gives on the contrary a result so much the less exact, all
other things being equal, as the points whose temperature we
are determining are more distant from the source.

The initial heat contained in a definite portion of the solid
mass penetrates successively the neighbouring parts, and spreads
itself in all directions; only an exceedingly small quantity of
it arrives at points whose distance from the origin is very great.

SECT. III.] HIGHEST TEMPERATURES IN A SOLID. 385

When we express analytically the temperature of these point?,
the object of the investigation is not to determine numerically
these temperatures, which are not measurable, but to ascertain
their ratios. Now these quantities depend certainly on the law
according to which the initial heat has been distributed, and the
effect of this initial distribution lasts so much the longer as the
parts of the prism are more distant from the source. But if the

terms which form part of the exponent, such as -rj- and -7-7-, have

4kt 4*kt

absolute values decreasing without limit, we may employ the
approximate integrals.

This condition occurs in problems where it is proposed to
determine the highest temperatures of points very distant from
the origin. We can demonstrate in fact that in this case the
values of the times increase in a greater ratio than the distances,
and are proportional to the squares of these distances, when the
points we are considering are very remote from the origin. It is
only after having established this proposition that we can effect
the reduction under the exponent. Problems of this kind are the
object of the following section.

SECTION III.

Of the highest temperatures in an infinite solid.

386. We shall consider in the first place the linear move
ment in an infinite bar, a portion of which has been uniformly
heated, and we shall investigate the value of the time which must
elapse in order that a given point of the line may attain its
highest temperature.

Let us denote by 2g the extent of the part heated, the middle
of which corresponds with the origin 0 of the distances x. All the
points whose distance from the axis of y is less than g and greater
than — g, have by hypothesis a common initial temperature f, and
all other sections have the initial temperature 0. We suppose
that no loss of heat occurs at the external surface of the prism, or,
which is the same thing, we assign to the section perpendicular to
the axis infinite dimensions. It is required to ascertain what will
F. H. 25

386 THEORY OF HEAT. [CHAP. IX.

be the time t which corresponds to the maximum of temperature
at a given point whose distance is x.

We have seen, in the preceding Articles, that the variable
temperature at any point is expressed by the equation

f«-*p

-FT-

The coefficient k represents -^n » ^ being the specific con-

Ox/

ducibility, C the capacity for heat, and D the density.

To simplify the investigation, make Jc = 1, and in the result

Tp»i

write kt or - instead of t. The expression for v becomes

7 72 J7

This is the integral of the equation -=- = -y- . The function -y-

cfa oar cfo;

measures the velocity with which the heat flows along the axis of
the prism. Now this value of -y-- is given in the actual problem
without any integral sign. We have in fact

a — x _!

p

or, effecting the integration,

^=_/_

dx 2

387. The function ~z may also be expressed without the

(ll)

sign of integration: now it is equal to a fluxion of the first order^-;
hence on equating to zero this value of -=- , which measures the

Uit

instantaneous increase of the temperature at any point, we have
the relation sought between x and t. We thus find

- 2 (* + ff~) f-^ , 2 (« -

~

SECT. III.] TIMES OF HIGHEST TEMPERATURES. 387

which gives

(x+V)* C*j^)2

(%+g}6~ ~v = (x-g}e~ « ;
whence we conclude

J£

We have supposed -rrf^ = \. To restore the coefficient we

Kt

must write -^ instead of t, and we have

__ffCD x

'~K~r

The highest temperatures follow each other according to the
law expressed by this equation. If we suppose it to represent the
varying motion of a body which describes a straight line, x being
the space passed over, and t the time elapsed, the velocity of
the moving body will be that of the maximum of temperature.

When the quantity g is infinitely small, that is to say when the
initial heat is collected into a single element situated at the

origin, the value of t is reduced to - , and by differentiation or

„ , Kt x*

development in series we find -^ = — .

(jD 2>

We have left out of consideration the quantity of heat which
escapes at the surface of the prism ; wTe now proceed to take account
of that loss, and we shall suppose the initial heat to be contained
in a single element of the infinite prismatic bar.

388. In the preceding problem we have determined the
variable state of an infinite prism a definite portion of which was
affected throughout with an initial temperature f. We suppose
that the initial heat was distributed through a finite space from
x = 0 to x = b.

We now suppose that the same quantity of heat If is contained
in an infinitely small element, from x = 0 to x = a). The tempera-

25—2

388 THEORY OF HEAT. [CHAP. IX.

ture of the heated layer will therefore be — , and from this follows

CO

what was said before, that the variable state of the solid is
expressed by the equation

fb e^t
-— J ~~ht (a) ;

this result holds when the coefficient -^ which enters into the

L/JJ

differential equation -=- = -^= -^—z — hv, is denoted by k. As to the

777

coefficient h, it is equal to /^rtc/; S denoting the area of the

section of the prism, I the contour of that section, and H the
conducibility of the external surface.

Substituting these values in the equation (a) we have

f represents the mean initial temperature, that is to say, that
which a single point would have if the initial heat were distributed
equally between the points of a portion of the bar whose length
is /, or more simply, unit of measure. It is required to determine
the value t of the time elapsed, which corresponds to a maximum
of temperature at a given point.

To solve this problem, it is sufficient to derive from equation
(a) the value of -7- , and equate it to zero ; we have
dv , x* lv

hence the value 0, of the time which must elapse in order that the
point situated at the distance x may attain its highest temperature,
is expressed by the equation

SECT. III.] VALUES OF HIGHEST TEMPERATURES. 389

To ascertain the highest temperature V, we remark that the
exponent of e~l in equation (a) is ht + -jy- • Now equation (&)

#2 1 xz x2 1 1

gives fa = jf- - ~ ; hence ht + 77- ; = ny-, - ~ , and putting for - its

"rA-C 2 db/JC Zfff 2 I

/p2 l\ £

known value, we have ht + TT~ , = \/ T + 7 ^2 ; substituting this ex-

j^rCv y T /(J

ponent of e"1 in equation (a), we have

and replacing */#& by its known value, we find, as the expression
of the maximum V,

'4/i 1 _1

X*

The equations (c) and (d) contain the solution of the problem ;

TTJ jr

let us replace h and k by their values TT/T^ and ^7^ ; let us also

-I Q

write 5 g instead of -=- , representing by g the semi-thickness of the

prism whose base is a square. We have to determine Fand 6,
the equations

we-

I*B ,l

V^^+i

These equations are applicable to the movement of heat in a
thin bar, whose length is very great. We suppose the middle of
this prism to have been affected by a certain quantity of heat bf
which is propagated to the ends, and scattered through the convex
surface. V denotes the maximum of temperature for the point
whose distance from the primitive source is a?; 0 is the time
which has elapsed since the beginning of the diffusion up to the
instant at which the highest temperature V occurs. The coeffi-

300 THEORY OF HEAT. [CHAP. IX.

cients C, H, K, D denote the same specific properties as in the
preceding problems, and g is the half-side of the square formed by
a section of the prism.

389. In order to make these results more intelligible by a
numerical application, we may suppose that the substance of which
the prism is formed is iron, and that the side 2g of the square is
the twenty-fifth part of a metre.

We measured formerly, by our experiments, the values of H
and K ; those of C and D were already known. Taking the metre
as the unit of length, and the sexagesimal minute as the unit of
time, and employing the approximate values of H, K} C, D, we
shall determine the values of V and 6 corresponding to a given
distance. For the examination of the results which we have in view,
it is not necessary to know these coefficients with great precision.

We see at first that if the distance x is about a metre and a

half or two metres, the term -^- #2, which enters under the radical,

Kg

has a large value with reference to the second term - . The ratio

of these terms increases as the distance increases.

Thus the law of the highest temperatures becomes more and
more simple, according as the heat removes from the origin. To
determine the regular law which is established through the whole
extent of the bar, we must suppose the distance x to be very
great, and we find

Kg

390. We see by the second equation that the time which corre
sponds to the maximum of temperature increases proportionally
with the distance. Thus the velocity of the wave (if however we
may apply this expression to the movement in question) is constant,
or rather it more and more tends to become so, and preserves this
property in its movement to infinity from the origin of heat.

SECT. III.] LAW OF THE HIGHEST TEMPERATURES. 391

We may remark also in the first equation that the quantity

JJH

fe~* K9 expresses the permanent temperatures which the
different points of the bar would take, if we affected the origin
with a fixed temperature /, as may be seen in Chapter I.,
Article 76.

In order to represent to ourselves the value of V, we must
therefore imagine that all the initial heat which the source con
tains is equally distributed through a portion of the bar whose
length is b, or the unit of measure. The temperature /, which
would result for each point of this portion, is in a manner the
mean temperature. If we supposed the layer situated at the
origin to be retained at a constant temperature /during an infinite
time, all the layers would acquire fixed temperatures whose

_ Jw •
general expression is fe K& , denoting by x the distance of the

layer. These, fixed temperatures represented by the ordinates of
a logarithmic curve are extremely small, when the distance is
considerable ; they decrease, as is known, very rapidly, according
as we remove from the origin.

Now the equation (8) shews that these fixed temperatures,
which are the highest each point can acquire, much exceed the
highest temperatures which follow each other during the diffusion
of heat. To determine the latter maximum, we must calculate
the value of the fixed maximum, multiply it by the constant

/2jy\i i
number ( ^- ) —j=- , and divide by the square root of the dis-

W' V^TT

tance x.

Thus the highest temperatures follow each other through the
whole extent of the line, as the ordinates of a logarithmic curve
divided by the square roots of the abscissae, and the movement of
the wave is uniform. According to this general law the heat
collected at a single point is propagated in direction of the length
of the solid.

391. If we regarded the conducibility of the external surface
of the prism as nothing, or if the conducibility K or the thickness
2g were supposed infinite, we should obtain very different results.

302 THEORY OF HEAT. [CHAP. IX.

We could then omit the term -=?- x~} and we should have1

K9

In this case the value of the maximum is inversely propor
tional to the distance. Thus the movement of the wave would
not be uniform. It must be remarked that this hypothesis is
purely theoretical, and if the conducibility H is not nothing, but
only an extremely small quantity, the velocity of the wave is not
variable in the parts of the prism which are very distant from the
origin. In fact, whatever be the value of Ht if this value is given,
as also those of K and g, and if we suppose that the distance x

211

increases without limit, the term -~r xz will always become much

&9
greater than J. The distances may at first be small enough for

2H

the term -=- #2 to be omitted under the radical. The times are
A#

then proportional to the squares of the distances ; but as the heat
flows in direction of the infinite length, the law of propagation
alters, and the times become proportional to the distances. The
initial law, that is to say, that which relates to points extremely
near. to the source, differs very much from the final law which is
established in the very distant parts, and up to infinity : but, in
the intermediate portions, the highest temperatures follow each
other according to a mixed law expressed by the two preceding
equations (D) and ((7),

392. It remains for us to determine the highest temperatures
for the case in which heat is propagated to infinity in every direc
tion within the material solid. This investigation, in accordance
with the principles which we have established, presents no
difficulty.

When a definite portion of an infinite solid has been heated,
and all other parts of the mass have the same initial temperature 0,
heat is propagated in all directions, and after a certain time the
state of the solid is the same as if the heat had been originally
collected in a single point at the origin of co-ordinates. The time

1 See equations (D) and (C), article 388, making 6 = 1. [A. F.]

SECT. III.] GENERAL INVESTIGATION. 393

which must elapse before this last effect is set up is exceedingly
great when the points of the mass are very distant from the origin.
Each of these points which had at first the temperature 0 is
imperceptibly heated; its temperature then acquires the greatest
value which it can receive; and it ends by diminishing more and
more, until there remains no sensible heat in the mass. The
variable state is in general represented by the equation

V=fdajdbfdoe- ^ -/(o,M ......... (E).

The integrals must be taken between the limits

The limits — alt + a2, — blt + b2, — c1, + c2 are given; they
include the whole portion of the solid which was originally heated.
The function f(a, b, c) is also given. It expresses the initial
temperature of a point whose co-ordinates are a, b, c. The defi
nite integrations make the variables a, b, c disappear, and there
remains for v a function of x, y, z, t and constants. To determine
the time 0 which corresponds to a maximum of v, at a given point

ra, we must derive from the preceding equation the value of -57:

at

we thus form an equation which contains 6 and the co-ordinates of
the point ra. From this we can then deduce the value of 6. If
then we substitute this value of 6 instead of t in equation (E), we
find the value of the highest temperature V expressed in x} y} z
and constants.

Instead of equation (E) let us write

v = (da fdb jdc Pf(a, b, c),
denoting by P the multiplier of f (a, b, c), we have

dt = ~2 t+jda db)dc gs

393. We must now apply the last expression to points of the
solid which are very distant from the origin. Any point what
ever of the portion which contains the initial heat, having for co
ordinates the variables a, b, c, and the co-ordinates of the point m

394 THEORY OF HEAT. [CHAP. IX.

whose temperature we wish to determine being x, y, z, the square of
the distance between these two points is (a — xf + (6 — y)*+ (c — z}2;

and this quantity enters as a factor into the second term of -7- .

Now the point m being very distant from the origin, it is
evident that the distance A from any point whatever of the heated
portion coincides with the distance D of the same point from the
origin ; that is to say, as the point m removes farther and farther
from the primitive source, which contains the origin of co-ordinates,
the final ratio of the distances D and A becomes 1.

It follows from this that in equation (e) which gives the value

of ^ the factor (a - xf + (b - yf + (c - zf- may be replaced by
dt

$ 4. y* + £ or r2, denoting by r the distance of the point m from
the origin. We have then

dv= /r^__3A
dt " V P 2 1) '

or

ai \ti ziy

If we put for v its value, and replace t by -^.t in order to

K'

re-establish the coefficient fTn w^ica we na(^ supposed equal to 1,

we have
dv

GD

394. This result belongs only to the points of the solid whose
distance from the origin is very great with respect to the greatest
dimension of the source. It must always be carefully noticed that
it does not follow from this condition that we can omit the varia
bles a, b, c under the exponential symbol. They ought only to be
omitted outside this symbol. In fact, the term which enters under
the signs of integration, and which multiplies / (a, 6, c), is the

SECT. III.] CONDITIONS FOR DISTANT POINTS. 395

product of several factors, such as

-a2 2 ax -x*

Now it is not sufficient for the ratio - to be always a very

great number in order that we may suppress the two first factors.
If, for example, we suppose a equal to a decimetre, and x equal to
ten metres, and if the substance in which the heat is propagated is
iron, we see that after nine or ten hours have elapsed, the factor

2 ax
7™ .

e CD is still greater than 2 ; hence by suppressing it we should

reduce the result sought to half its value. Thus the value of -r- ,

dt

as it belongs to points very distant from the origin, and for any
time whatever, ought to be expressed by equation (a). But it is
not the same if we consider only extremely large values of the
time, which increase in proportion to the squares of the distances :
in accordance with this condition we must omit, even under the
exponential symbol, the terms which contain a, b, or c. Now this
condition holds when we wish to determine the highest tempera
ture which a distant point can acquire, as we proceed to prove.

395. The value of ^- must in fact be nothing in the case in
question ; we have therefore

Thus the time which must elapse in order that a very distant
point may acquire its highest temperature is proportional to the
square of the distance of this point from the origin.

If in the expression for v we replace the denominator -^=-

VjU

2
by its value r2, the exponent of e~l which is

396 THEOKY OF HEAT. [CHAP. IX.

may be reduced to ~ , since the factors which we omit coincide with

L

unity. Consequently we find
«V

V =

The integral Ida Idb ldcf(a, b, c) represents the quantity of

the initial heat : the volume of the sphere whose radius is r is

4

K 7rrs, so that denoting by / the temperature which each molecule
o

of this sphere would receive, if we distributed amongst its parts
all the initial heat, we shall have v = A/ — $f.

The results which we have developed in this chapter indicate
the law according to which the heat contained in a definite portion
of an infinite solid progressively penetrates all the other parts
whose initial temperature was nothing. This problem is solved
more simply than that of the preceding Chapters, since by
attributing to the solid infinite dimensions, we make the con
ditions relative to the surface disappear, and the chief difficulty
consists in the employment of those conditions. The general
results of the movement of heat in a boundless solid mass are
very remarkable, since the movement is not disturbed by the
obstacle of surfaces. It is accomplished freely by means of the
natural properties of heat. This investigation is, properly
speaking, that of the irradiation of heat within the material
solid.

SECTION IV.

Comparison of the integrals.

396. The integral of the equation of the propagation of heat
presents itself under different forms, which it is necessary to com
pare. It is easy, as we have seen in the second section of this
Chapter, Articles 372 and 376, to refer the case of three dimen
sions to that of the linear movement ; it is sufficient therefore to
integrate the equation

** JL'&*

dt~~ CDdx*'

SECT. IV.] FORM OF THE INTEGRAL FOR A RING. 397

or the equation

dv d?v

To deduce from this differential equation the laws of the propa
gation of heat in a body of definite form, in a ring for example,
it was necessary to know the integral, and to obtain it under a
certain form suitable to the problem, a condition which could be
fulfilled by no other form. This integral was given for the first
time in our Memoir sent to the Institute of France on the
21st of December, 1807 (page 124, Art. 84) : it consists in the
following equation, which expresses the variable system of tem
peratures of a solid ring :

/.

(a).

R is the radius of the mean circumference of the ring ; the integral
with respect to a. must be taken from a = 0 to a. = ZnR, or, which
gives the same result, from a = — irR to a = TrR ; i is any integer,
and the sum 2) must be taken from i = — oo to i= + x ; v denotes
the temperature which would be observed after the lapse of a
time t, at each point of a section separated by the arc x from that
which is at the origin. We represent by v = F (x) the initial tem
perature at any point of the ring. We must give to i the succes
sive values

0, +1, +2, +3, &c., and -1, -2, - 35 &c.,

and instead of cos — — 0 — ' write
M

ix IOL . ix . la.

We thus obtain all the terms of the value of v. Such is the
form under which the integral of equation (a) must be placed, in
order to express the variable movement of heat in a ring (Chap. IV.,
Art. 241). We consider the case in which the form and extent of
the generating section of the ring are such, that the points of the
same section sustain temperatures sensibly equal. We suppose
also that no loss of heat occurs at the surface of the ring.

398 THEORY OF HEAT. [CHAP. IX.

397. The equation (a) being applicable to all values of R, we
can suppose in it R infinite ; in which case it gives the solution of
the following problem. The initial state of a solid prism of
small thickness and of infinite length, being known and expressed
by v — F(x)t to determine all the subsequent states. Consider the
radius E to contain numerically n times the unit radius of the
trigonometrical tables. Denoting by q a variable which successively
becomes dq, 2dq, 3dq, ... idq, &c., the infinite number n may

be expressed by -y- , and the variable number i by -|- . Making
these substitutions we find

v = ^- £ dq I dy. F (a) e~qH cos q (x — a).

The terms which enter under the sign 2 are differential quan
tities, so that the sign becomes that of a definite integral ; and
we have

-j f'+ao M-ao

v = x- doL F (a) I dq e-& cos (qx - qz) (@).

J->7T J -oo J - oo

This equation is a second form of the integral of the equation
(QL) ; it expresses the linear movement of heat in a prism of infinite
length (Chap. VII., Art. 354). It is an evident consequence of the
first integral (a).

398. We can in equation (/3) effect the definite integration
with respect to q-} for we have, according to a known lemma, which
we have already proved (Art. 375),

/.

I

J

+00

dz e~z* cos 2hz = e~h*

-00

Making then z* = (ft, we find

» Jt

Hence the integral (/S) of the preceding Article becomes

r

J

J -

SECT, iv.] LAPLACE'S FORM OF THE INTEGRAL. 399

If we employ instead of a another unknown quantity ft

making — = ft we find

%Jt

(8).

This form (8) of the integral l of equation (a) was given in
Volume VIII. of the Memoires de VEcole Poly technique, by M.Laplace,
who arrived at this result by considering the infinite series which
represents the integral.

Each of the equations (/3), (7), (8) expresses the linear diffusion
of heat in a prism of infinite length. It is evident that these are
three forms of the same integral, and that not one can be con
sidered more general than the others. Each of them is contained
in the integral (a) from which it is derived, by giving to R an
infinite value.

infm

r s

399. It is easy to develope the value of v deduced from
equation (a) in series arranged according to the increasing powers
of one or other variable. These developments are self-evident,
and we might dispense with referring to them; but they give rise
to remarks useful in the investigation of integrals. Denoting by

<j>, <£", (f>", &c., the functions -7- <£(#), -j— 2 $(#")> ~T~3 $(x}> &c-> we
have i

dv /, , r 7 // T~"* » \^

-77 = v , and v = c + 1 at v ;
1 A direct proof of the equivalence of the forms

tt

t —
F<f> (x + 2/3^) and e dic2 $ (x), (see Art. 401),

has been given by Mr Glaisher in the Messenger of Mathematics, June 1876, p. 30.
Expanding <(>(x+2p«Jt) by Taylor's Theorem, integrate each term separately:
terms involving uneven powers of >Jt vanish, and we have the second form ;
which is therefore equivalent to

]_ /•*> [3

~ I da I

T y-» Jo

from which the first form may be derived as above. We have thus a slightly
generalized form of Fourier's Theorem, p. 351. [A. F.]

400 THEORY OF HEAT. [CHAP. IX.

here the constant represents any function of x. Putting for v" its
value c" + ldtviv, and continuing always similar substitutions, we
find

v = c+ jdt v"

\c" +jdt (civ +jdt v«J] ,

or v = c + tc"+~dv + ^G« + ^c™ + &c ............. (I7).

In this series, c denotes an arbitrary function of x. If we wish
to arrange the development of the value of v, according to ascend
ing powers of #, we employ

d*v _ dv
dx*~dt'

and, denoting by <£y, <£/y, <£//y, &c. the functions
d, d* d*

a*' a?* df^ &c->

we have first v = a + bx + \dx \dx vt ; a and b here represent any
two functions of t. We can then put for v its value

a, + l>p + Idx Idx v/f ;

and for vti its value atl + b^x + Idx Idx v4llt and so on. By continued
substitutions

v= a + bx + \dx Idx vt
= a + lx+ \dx\dx \at + Ix 4- Idx Idx v J
= a + bx+ldx \dx at + bx + Idx Idx (au + btx -f \dx \dx v\ |

SECT, IV.] NUMBER OF ARBITRARY FUNCTIONS. 401

or t; = a + -^ at + r-r a 4- — a + &c.

|2 - |4 ' 6

l a .................. (Z).

O O

In this series, a and b denote two arbitrary functions of t.

If in, the series given by equation (^) we put, instead of
a and b, two functions </> (t) and -^ (f), and develope them according
to ascending powers of t, we find only a single arbitrary function
of x, instead of two functions a and b. We owe this remark to
M. Poisson, who has given it in Volume vi. of the Memoires de
TEcole Polytechnique, page 110.

Reciprocally, if in the series expressed by equation (T) we de
velope the function c according to powers of x, arranging the
result with respect to the same powers of x, the coefficients of
these powers are formed of two entirely arbitrary functions of t ;
which can be easily verified on making the investigation.

400. The value of v, developed according to powers of t,
ought in fact to contain only one arbitrary function of x ; for the
differential equation (a) shews clearly that, if we knew, as a
function of #, the value of v which corresponds to t = 0, the
other values of the function v which correspond to subsequent
values of t, would be determined by this value.

It is no less evident that the function v, when developed
according to ascending powers of x, ought to contain jwo com
pletely arbitrary functions of the variable t. In fact the dISerential

equation -7-3 = -7- shews that, if we knew as a function of t the

value of v which corresponds to a definite value of x, we could
not conclude from it the values of v which correspond to all the
other values of x. It would be necessary in addition, to give as
a function of t the value of v which corresponds to a second value
of x} for example, to that which is infinitely near to the first. All
the other states of the function v, that is to say those which corre
spond to all the other values of x, would then be determined. The
differential equation (a) belongs to a curved surface, the vertical
ordinate of any point being v, and the two horizontal co-ordinates
F. H. 26

402 THEORY OF HEAT. [CHAP. IX.

x and and t. It follows evidently from this equation (a) that the
form of the surface is determined, when we give the form of the
vertical section in the plane which passes through the axis of x :
and this follows also from the physical nature of the problem ; for
it is evident that, the initial state of the prism being given, all the
subsequent states are determined. But we could not construct
the surface, if it were only subject tcT passing through a curve
traced on the first vertical plane of i and v. It would be necessary
to know further the curve traced on a second vertical plane
parallel to the first, to which it may be supposed extremely near.
The same remarks apply to all partial differential equations, and
we see that the order of the equation does not determine in all
cases the number of the arbitrary functions.

401. The series (T) of Article 399, which is derived from the
equation

dv d?v

may be put under the form v = etD<i <f> (x). Developing the ex-

d*
ponential according to powers of D, and writing -j-. instead of D\

considering i as the order of the differentiation, we have

Following the same notation, the first part of the series (X)
(Art. 399), which contains only even powers of x, may be expressed
under the form cos (x ,J— D) <j> (t). Develope according to powers

of x, and write ^ instead of D\ considering i as the order of the

differentiation. The second part of the series (X) can be derived
from the first by integrating with respect to x, and changing the
function <£ (t) into another arbitrary function ty (t). We have
therefore

v = cos (tf^- !>)</>(£)+ W
and -W = I *dx cos (x J^

SECT. IV.] SYMBOLICAL METHODS. 403

This known abridged notation is derived from the analogy
which exists between integrals and powers. As to the use made
of it here, the object is to express series, and to verify them
without any development. It is sufficient to differentiate under
the signs which the notation employs. For example, from the
equation v = etl}* <f) (a?), we deduce, by differentiation with respect
to t only,

which shews directly that the series satisfies the differential
equation (a). Similarly, if we consider the first part of the series
(X), writing

we have, differentiating twice with respect to x only,

Hence this value of v satisfies the differential equation (a).

We should find in the same manner that the differential
equation

gives as the expression for v in a series developed according to
increasing powers of y,

v — cos (yD) $ (x).

We must develope with respect to y, and write ^- instead of
D : from this value of v we deduce in fact,

? = --D'COS

The value sin (yD} ty (x) satisfies also the differential equation;
hence the general value of v is

v = cos (yD) <£ (x) + W, where W= sin (yD) ty (x).

26—2

404 THEORY OF HEAT. [CHAP. IX.

402. If the proposed differential equation is

ifv dh efo / v

dt*-dtf + d?/z"

and if we wish to express v in a series arranged according to
powers of t, we may denote by D<£ the function

S?* + ^»*;

fPv
and the equation being -^ = Dv, we have

v = cos (t <J— D) $ (x, y).
From this we infer that

-Ta -5-5 -7-5 •

at dx* df

We must develope the preceding value of v according to powers
of t, write (n + -rni , instead of D', and then regard i as the order

n

of differentiation.

The following value \dt cos (t J- D) ^ (a?, #) satisfies the same
condition; thus the most general value of v is

jdt cos (* 7^5) ^ (x, y] ;

and

v is a function f(xy y> f) of three variables. If we make t = 0, we
have/= (a?, y, 0) = <£ (a?, y) ; and denoting ^/fe y, 0 by/ («, y, <),
we have/ (a?, y, 0) = ^ (x, y}.
If the proposed equation is

the value of v in a series arranged according to powers of t will

SECT. IV.] A DIFFERENTIAL EQUATION. 405

be v = cos (tD*) <j> (#,#), denoting ^ by D; for we deduce from
this value

d*

572 — -V = - -j-4 V.

dt dy?

The general value of v, which can contain only two arbitrary
functions of x and y, is therefore

v = cos (ZD2) 0 (a?, y) + W,

and TF = f dt cos (*Z>2) -^ (#, y).

Jo

Denoting u by /(a?, y, 0, and ^ by /' (a;, y, «), we have to
determine the two arbitrary functions,

* & y) =/ (^ y* °)> and ^ (^ y) =/ te y> o).

403. If the proposed differential equation is
tfv d*v d*v

_
-

we may denote by D$ the function -y + -gj so that

or Z)2^> can be formed by raising the binomial ( -j-a + -p 2 j to the

second degree, and regarding the exponents as orders of differen-

d?v
tiation. Equation (e) then becomes -^ + Dzv = 0; and the value

of v, arranged according to powers of t, is cos (tD) <f> (x, y) ; for
from this we derive

7 .„ ^ «/, or ^^ + -y-4 + 2 , 2 , 2 + -7-4 = 0.

ar cfo cfar dx dy dy

The most general value of v being able to contain only two
arbitrary functions of x and ?/, which is an evident consequence of
the form of the equation, may be expressed thus :

v = cos (tD) </> (x, y) + 1 dt cos (tD} f (#, y).

406 THEOKY OF HEAT. [CHAP. IX.

The functions </> and i/r are determined as follows, denoting the
function v by /(a?, y, t), and ^/ (x, y, t) by/ (x, y, t),

$ (*, y} =f (*, y> o), t fa y) =/x fa y. o).

Lastly, let the proposed differential equation be

dv

_- = a 1-2 -y~4 c :r~6
dt dot? dx* dx6

the coefficients a, b, c> d are known numbers, and the order of the
equation is indefinite.

The most general value of v can only contain one arbitrary
function of x ; for it is evident, from the very form of the equa
tion, that if we knew, as a function of x, the value of v which
corresponds to t — 0, all the other values of v, which correspond to
successive values of tt would be determined. To express v, we
should have therefore the equation v —etj)^ (x).

We denote by D(f> the expression

that is to say, in order to form the value of v, we must develop
according to powers of t, the quantity

a.* + ca6 + da.8 + &C.)

and then write -^- instead of a, considering the powers of a as orders
dx

of differentiation. In fact, this value of v being differentiated
with respect to t only, we have

dv detD , N _. d*v , d*v d*v p
-T: = ^r 9 (®) = -Dv = a -j—t + b -,— 4 + c -j—6 + &c.
c?^ ai «fic2 dx* da?

It would be useless to multiply applications of the same process.
For very simple equations we can dispense with abridged expres
sions ; but in general they supply the place of very complex in
vestigations. We have chosen, as examples, the preceding equa
tions, because they all relate to physical phenomena whose analytical
expression is analogous to that of the movement of heat. The two
first, (a) and (b), belong to the theory of heat ; and the three

SECT. IV.] OTHER MODES OF INTEGRATION. 407

following (c), (d), (e), to dynamical problems; the last (/) ex
presses what the movement of heat would be in solid bodies, if
the instantaneous transmission were not limited to an extremely
small distance. We have an example of this kind of problem in
the movement of luminous heat which penetrates diaphanous
media.

404. We can obtain by different means the integrals of these
equations : we shall indicate in the first place that which results
from the use of the theorem enunciated in Art. 361, which we
now proceed to recal.

If we consider the expression

/•+«> /•+<» p

dy. $ (a) I d<£

J - oo J -co

cos (px-pz), .................. (a)

we see that it represents a function of #; for the two definite
integrations with respect to a and p make these variables dis
appear, and a function of x remains. Thgjiataiir of the function
will evidently depend on that which we shall have chosen for
(j) (a). We may ask what the function <f) (a), ought to be, in order
tffSTafter two definite integrations we may obtain a given function
f(x^. In general the investigation of the integrals suitable for
the expression of different physical phenomena, is reducible to
problems similar to the preceding. The object of these problems
is to determine the arbitrary functions under the signs of the
definite integration, so that the result of this integration may be
a given function. It is easy to see, for example, that the general
integral of the equation

dv d*v d4v dev d*v - ,

would be known if, in the preceding expression («), we could
determine <£ (a), so that the result of the eq^kion might be a
given function f (x). In fact, we form directly a particular value
of v, expressed thus,

v = e~mt cospx,

and we find this condition,

m = op* -f lp* + rp6 + &c.

408 THEORY OF HEAT. [CHAP. IX.

We might then also take

v _ e-mt cos

giving to the constant a any value. We have similarly
v**fd*<j> 0) e-*(^+6*4+^6+&c) cos (px -pz).

It is evident that this value of v satisfies the differential equation
(/) ; it is merely the sum of particular values.

Further, supposing t = 0, we ought to find for v an arbitrary
function of x. Denoting this function by/(#), we have

/ (x) = I dz (f> (a) I dp cos (px — p%).

Now it follows from the form of the equation (/), that the most
general value of v can contain only one arbitrary function of x.
In fact, this equation shews clearly that if we know as a function
of x the value of v for a given value of the time t, all the other
values of v which correspond to other values of the time, are
necessarily determined. It follows rigorously that if we know,
as a function of t and x, a value of v which satisfies the differential
equation; and if further, on making t = 0, this function of x and t
becomes an entirely arbitrary function of x, the function of x and
t in question is the general integral of equation (/). The whole
problem is therefore reduced to determining, in the equation
above, the function <£ (a), so that the result of two integrations
may be a given function /(#). It is only necessary, in order that
the solution may be general, that we should be able to take for
f(x) an entirely arbitrary and even discontinuous function. It is
merely required therefore to know the relation which must always
exist between the given function f(x) and the unknown function
<j> (a). Now this very simple relation is expressed by the theorem
of which we are speaking. It consists in the fact that when the
integrals are taken between infinite limits, the function <£ (a) is

~— / (a) ; that is to say, that we have the equation

I r+oo /•+

~fc.-l &/(a)|

^?r j - oo j -

SECT. IV.] VIBRATION OF ELASTIC LAMINA. 409

From this we conclude as the general integral of the proposed
equation (/),

u = -L [ efe/(

^7T J -oo '

405. If we propose the equation

which expresses the transverse vibratory movement of an elastic
plate1, we must consider that, from the form of this equation, the
most general value of v can contain only two arbitrary functions
of x: for, denoting this value of v by f(x,t), and the function

-rf(x, t) by /' (a?, t), it is evident that if we knew f(x, 0) and
cit

f (x, 0), that is to say, the values of v and - - at the first instant,

at

all the other values of v would be determined.

This follows also from the very nature of the phenomenon. In
fact, consider a rectilinear elastic lamina in its state of rest: x is
the distance of any point of this plate from the origin of co
ordinates; the form of the lamina is very slightly changed, by
drawing it from its position of equilibrium, in which it coincided
with the axis of x on the horizontal plane; it is then abandoned to
its own forces excited by the change of form. The displacement is
supposed to be arbitrary, but very small, and such that the initial
form given to the lamina is that of a curve drawn on a vertical
plane which passes through the axis of x. The system will suc
cessively change its form, and will continue to move in the vertical
plane on one side or other of the line of equilibrium. The most
general condition of this motion is expressed by the equation

d*v d4v ,, . ,

a?+»-° ........................ w-

Any point m, situated in the position of equilibrium at a
distance x from the origin 0, and on the horizontal plane, has, at

1 An investigation of the general equation for the lateral vibration of a thin
elastic rod, of which (d) is a particular case corresponding to no permanent
internal tension, the angular motions of a section of the rod being also neglected,
will be found in Donkiu's Acoustics, Chap. ix. §§ 169—177. [A.F.]

410 THEORY OF HEAT. [CHAP. IX.

the end of the time £, been removed from its place through the
perpendicular height v. This variable flight v is a function of
x and t. The initial value of v is arbitrary; it is expressed by any
function (/> (x). Now, the equation (d) deduced from the funda
mental principles of dynamics shews that the second fluxion

of v, taken with respect to £,-or ~z , and the fluxion of the fourth

(Jut

d*v
order taken with respect to x, or ^ 4 are two functions of x and t,

which differ only in sign. We do not enter here into the special
question relative to the discontinuity of these functions; we have
in view only the analytical expression of the integral.

We may suppose also, that after having arbitrarily displaced
the different points of the lamina, we impress upon them very
small initial velocities, in the vertical plane in which the vibrations
ought to be accomplished. The initial velocity given to any
point m has an arbitrary value. It is expressed by any function
ty (x} of the distance x.

It is evident that if we have given the initial form of the
system or </> (x) and the initial impulses or ty (x), all the subse
quent states of the system are determinate. Thus the function
v oif(x,t), which represents, after anytime t, the corresponding
form of the lamina, contains two arbitrary functions <£ (x)
and ijr (x).

To determine the function sought f(xt t), consider that in the
equation

we can give to v the very simple value
u — cos (ft cos qXj
or else u — cos ft cos (qx — <?a) ;

denoting by q and a any quantities which contain neither x nor t.
We therefore also have

u = I doL F(OL) Idq cos ft cos (qx — q1*),

SECT. IV.] SOLUTION OF EQUATION OF VIBRATION. 411

F(OL) being any function, whatever the limits of the integrations
may be. This value of v is merely a sum of particular values.

Supposing now that t = 0, the value of v must necessarily
be that which we have denoted by/(#, 0) or <f> (x). We have
therefore

(f) (x) = IdoL F (a) \dq cos (qx — qx).

The function F (a) must be determined so that, when the two \
integrations have been effected, the result shall be the arbitrary I j
function <j> (x). Now the theorem expressed by equation (.6) shews J
that when the limits of both integrals are — oo and + GO , we A
have

Hence the value of u is given by the following equation :

I /•+» [+*>

u = ^— dy. </> (a) I dq cos ft cos (qx — qa).

Air J -so J -oo

If this value of u were integrated with respect to t, the <£ in
it being changed to ^Jr, it is evident that the integral (denoted
by W) would again satisfy the proposed differential equation (d),
and we should have

W=27rjd*^ W fa -2 sin & cos (<lx - 2*)-

This value W becomes nothing when £ = 0; and if we take the
expression

dw

dw i r+

"^ = 2^rJ

we see that on making t — 0 in it, it becomes equal to -^ (x).
The same is not the case with the expression -j- ; it becomes -
nothing when t = 0, and u becomes equal to <£ (x) when t = 0.

It follows from this that the integral of equation (d) is

1 r+x r+ao

# = —I da<t>(a) \ dq cos ^^ cos (qx — qz) + W= u + TF,

^7T J -oo J - x

and

1 .

j Sin Q"t COS (QX —

i r"1"00 r+ao 1

Tr= g- I rfaA|r (a) I dq —^

412 THEORY OF HEAT. [CHAP. IX.

In fact, this value of v satisfies the differential equation (d) ;
also when we make t — 0, it becomes equal to the entirely arbitrary

function fy (x) ; and when we make t = 0 in the expression -7- ,

cLii

it reduces to a second arbitrary function ^r (as). Hence the value
of v is the complete integral of the proposed equation, and there
cannot be a more general integral.

406. The value of v may be reduced to a simpler form by
effecting the integration with respect to q. This reduction, and
that of other expressions of the same kind, depends on the two
results expressed by equations (1) and (2), which will be proved
in the following Article.

dq cos ^ cos qz =— p-sin I-T + T) (1).

* * * v '

/:

Ciq sin q*t cos qz — — .-= sin f-r — -T- ) (2).
-*••*• -^ /.£ \ ZL AiT1 / \ /

» (k/« \™ **/

From this we conclude

Denoting — j- by another unknown p, we have

a = x + 2/,6 Jt, da. =
Putting in place of sin (^ +Afc2J its value

1

v
we have

u = -TT= f ^ (sin ^2 + cos fS) <j> (OL + 2/4 V/) ........ («').

V ^7T J -oo

We have proved in a special memoir that (5) or (8'), the
integrals of equation (d), represent clearly and completely the
motion of the different parts of an infinite elastic lamina. They
contain the distinct expression of the phenomenon, and readily
explain all its laws. It is from this point of view chiefly that we

SECT. IV.] TWO DEFINITE INTEGRALS. 413

have proposed them to the attention of geometers. They shew
how oscillations are propagated and set up through the whole
extent of the lamina, and how the effect of the initial displace
ment, which is arbitrary and fortuitous, alters more and more as
it recedes from the origin, soon becoming insensible, and leaving
only the existence of the action of forces proper to the system, the
forces namely of elasticity.

407. The results expressed by equations (1) and (2) depend
upon the definite integrals

I dx cos ce2, an d I dx sin x* ;

f-f-oo • r-f-oo

g = I dx cos cc2, and h = I dx sin a;2;

J — ao J - ao

let

and regard g and h as known numbers. It is evident that in the
two preceding equations we may put y + b instead of x, denoting
by b any constant whatever, and the limits of the integral will be
the same. Thus we have

g = P*dy cos (y* + Zby + b2), h = ( ^ dy sin (y2 + 2by + 62),

J — 00 J —00

= f di I cos ^ cos 2^ cos ^ ~~ cos ^2 s*

J I— sin y2 sin 2by cos 68 - sin y2 cos 2by sin b2)

Now it is easy to see that all the integrals which contain the
factor sin 2by are nothing, if the limits are — <x> and + o> ; for
sin 2by changes sign at the same time as y. We have therefore

g = cos 6a I dy cos yz cos 2by - sin b* I dy sin y* cos 2by ......... (a).

The equation in h also gives

h = id i S^n y* cos 2^ cos ^ + cos y* cos ^y sm

J \ + cos y2 sin 2by cos b2 — sin y2 sin 26y sin
and, omitting also the terms which contain sin 2by, we have

h - cos &2 J dy sin y2 cos 2by + sin Z>2 / dy cos 2/2 cos 2by ........ (6).

414 THEORY OF HEAT. [CHAP. IX.

The two equations (a) and (b) give therefore for g and h the
two integrals

\dy sin yz cos 2&# and \dy cos ?/2 cos 2%

which we shall denote respectively by A and B. We may now
make

sn cos >2 =

yz = pzt, and Zby = pz ; or i
we have therefore

fj"t\dp cosp*t cos£)2 = A, *Jt\dp si]

The values1 of g and /& are derived immediately from the known
result

r + oo

VTT = I dx e~x*.

J -00

The last equation is in fact an identity, and consequently does
not cease to be so, when we substitute for # the quantity

The substitution gives

= r1 \dy e'"^= f1 \dy

Thus the real part of the second member of the last equation
is N/TT and the imaginary part nothing. Whence we conclude

N/TT = -j= (\dy cos y*+jdy sin y*) ,

1 More readily from the known results given in § 360, viz. —

fdusinu /^ , du . ..

~~r~ = \/ o • Let u = z> % 1= =dz> then
x/w v 2 Ju

I e?2sins2=i \/ J. and I dzsinz*=2 I dzsiuz"*= \/ J.

Jo V 2 J-oo Jo V 2

So for the cosine from p**^« /* [B.L.B.]

/w ^ 2

SECT. IV.] VALUES OF THE INTEGRALS. 415

and 0 = \dy cos y* — \dy siny2,

or

It remains only to determine, by means of the equations (a)
and (6), the values of the two integrals

I dy cos yz cos 2by and | dy sin y* sin 2by.
They can be expressed thus :

A = I dy cos y* cos 2by = h sin 62 + g cos 52,

B = I dy sin i/2 cos 26^ = h cos 62- ^ sin b2 ;
whence we conclude

writing sin ^ , or cos ^ instead of i/ - , we have

= ^sing4-|) .................. (1)

and I dpsmtft cospz= -ILsmt7^—^-}.., ,..(2)

/- 4 kt)

408. The proposition expressed by equation (B) Article 404,
or by equation (E) Article 361, which has served to discover the
integral (8) and the preceding integrals, is evidently applicable to
a very great number of variables. In fact, in the general equation

J /•+*> -+QO

or / 0) = 9- / dP\ d* cos (px - p*)f (a),

A- V< . — OO J ~ 3D

41 G THEORY OF HEAT. [CHAP. IX.

we can regard f(x) as a function of the two variables x and y.
The function /(a) will then be a function of a and y. We shall
now regard this function f (a, y) as a function of the variable y,
and we then conclude from the same theorem (B), Article 404,

1 f"1"00 f

that f(a, ;?/) = — J^ / (a, £) jdq cos (qy -

We have therefore, for the purpose of expressing any function
whatever of the two variables x and ?/, the following equation

y) = **&f(«' $ cos (P*-

/•+oo

J -00

We form in the same manner the equation which belongs to
functions of three variables, namely,

*, y, *) = ** A 7)

jd/p cos (_p# — |>a) /Jg cos (^ - 0/9) I«?r cos (r« — ry) ..... (BBF),

each of the integrals being taken between the limits — oo
and

It is evident that the same proposition extends to functions
which include any number whatever of variables. It remains to
show how this proportion is applicable to the discovery of the
integrals of equations which contain more than two variables.

409. For example, the differential equation being

we wish to ascertain the value of v as a function of (x, y, t), such
that ; 1st, on supposing t = 0, v or f(x, y, t) becomes an arbitrary
function <£ (a?, y) of x and y\ 2nd, on making t = 0 in the value

S/'IJ

of --y or f (x,yy t), we find a second entirely arbitrary function

SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 417

From the form of the differential equation (c) we can infer
that the value of v which satisfies this equation and the two pre
ceding conditions is necessarily the general integral. To discover
this integral, we first give to v the particular value

v = cos mt cos px cos qy.

The substitution of v gives the condition m = Jp* + q*.
It is no less evident that we may write

v = cosp (x — a) cos q (y — ft) cos t J$ -f (f,

or

v = I dx I d/3 F (a, /3) I dp cos (px - pot) Idq cos (qy - q@) cos t Jp* + q*t

whatever be the quantities p, q, a, ft and F (a, @), which contain
neither x, y, nor t. In fact this value of t is merely the sum of
particular values.

If we suppose t = 0, v necessarily becomes $ (x} y). "We have
therefore

0 (x> y) = jdzldP F (a, /3) J dp cos (px - POL) jdq cos (qy - q/3).

Thus the problem is reduced to determining F (a, /3), so that
the result of the indicated integrations may be <£ (x, y). Now, on
comparing the last equation with equation (BB), we find

*> y} = ( *-} f «k f + ^ </> (a> /S) f +

\^7r/ J-ao J-x> J -

cos -

£0
0

Hence the integral may be expressed thus :

We thus obtain a first part i« of the integral; and, denoting
by W the second part, which ought to contain the other arbitrary
function i/r (x, y), we have

v = u+ W,
F. H. 27

418 THEORY OF HEAT. [CHAP. IX.

and we must take W to be the integral ludt, changing only <£>

into A/T. In fact, u becomes equal to (f> (a?, y), when t is made
= 0; and at the same time W becomes nothing, since the integra
tion, with respect to t, changes the cosine into a sine.

Further, if we take the value of -7-, and make t = 0, the first

part, which then contains a sine, becomes nothing, and the
second part becomes equal to ty (x, y). Thus the equation
v = u+ Wis the complete integral of the proposed equation.

We could form in the same manner the integral of the
equation

<Fv c?v d?v cFv

It would be sufficient to introduce a new factor

2^ cos (rz - ry) ,
and to integrate with respect to r and 7.

410. Let the proposed equation be ;r^ + -7-2 + -7-* — $ ; it is

ctx cLy ctz

required to express v as a function f(x,y,z), such that, 1st,
f(x,y,Q) may be an arbitrary function $(#,#); 2nd, that on

making 2 = 0 in the function -7- f(x,y,z) we may find a second

ctz

arbitrary function ^ (#, y). It evidently follows, from the form of
the differential equation, that the function thus determined will
be the complete integral of the proposed equation.

To discover this equation we may remark first that the equa
tion is satisfied by writing v = cos^>#cos qij emz, the exponents
p and q being any numbers whatever, and the value of m being

We might then also write

v = cos (px-p*} cos (qy - q(3} (e<v^+«i -f

SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 419

or

t' =

t, ft) jdpjdq cos (px -pi) cos (qy - qft)

If 2 be made equal to 0, we have, to determine F(y, /3), the
following condition

0 (•»» y) = jdzldft F (a, /3) jdpjdq cos (^ -_pa) cos (^ - qft) ;
and, on comparing with the equation (BB)t we see that

we have then, as the expression of the first part of the integral,
» ^) 4P cos (Px -P*)d(l cos (^ ~ 2#)

The value of w reduces to </> (x, y) when « = 0, and the same

substitution makes the value of -j- nothing.

dx

We might also integrate the value of u with respect to z, and
give to the integral the following form in which i/r is a new
arbitrary function:

IF= ^) | da. jd/3 ^r (a, ft) Jdp cos (^ - pa) jdq cos (jy - qft)

The value of TF becomes nothing when « = 0, and the same

dW
substitution makes the function —j~ equal to -^ (x, y). Hence

the general integral of the proposed equation is v = u + W.

411. Lastly, let the equation be

f
dt

* ' *-~*^dy*~

27—2

420 THEORY OF HEAT. [CHAP. IX.

it is required to determine v as a function/ (#, y, t), which satisfies
the proposed equation (e) and the two following conditions :
namely, 1st, the substitution t — 0 in f(x,yji) must give an
arbitrary function <jf> (x, y) ; 2nd, the same substitution in

— f (x-, y, t) must give a second arbitrary function ty (x, y).
ctt

It evidently follows from the form of equation (e), and from
the principles which we have explained above, that the function v,
when determined so as to satisfy the preceding conditions, will be
the complete integral of the proposed equation. To discover this
function we write first,

v = cos px cos qy cos mt,
whence we derive

d*v 2 d*v 4 d*» 22 d'v

= -.m*v = tfv = p*g*v =^

dt dx* dor dy* * • dy*

We have then the condition m=p* + q*. Thus we can write

v = cospx cos qy cos t (p* + £a),
or v = cos (px —px) cos (qy — q/3) cos (p*t -1- q*t),

or v = ldz \dpF(z, j3) Idp \dq cos (px — pot) cos (qy — q/5)

cos (pzt + q*t).

When we make t = 0, we must have v = <f>(x,y)\ which serves
to determine the function F (a., /9). If we compare this with the
general equation (BB), we find that, when the integrals are taken

/ 1 \2
between infinite limits, the value of F(a, ft) is I — \ (f> (a, /8). We

have therefore, as the expression of the first part u of the
integral,

J

u = a' cos ~a cos ~

Integrating the value of w with respect to t, the second arbi
trary function being denoted by -\|r, we shall find the other part
W of the integral to be expressed thus :

SECT. IV.] OTHER FORM OF INTEGRAL. 421

W = (^) fa fa ^ (*, ft) fa fa COS (px -jpa) COS fe/ - 2/3)

sin (jp'l + g*t)

If we make t = 0 in u and in IF, the first function becomes
equal to $(&,y), and the second nothing; and if we also make

£ = 0 in -j-u and in -=- W, the first function becomes nothing,

and the second becomes equal to ty (x,y)' hence v = u + W is the
general integral of the proposed equation.

412. We may give to the value of u a simpler form by effect
ing the two integrations with respect to p and q. For this
purpose we use the two equations (1) and (2) which we have
proved in Art. 407, and we obtain the following integral,

Denoting by u the first part of the integral, and by W the
second, which ought to contain another arbitrary function, we
have

rt
TF =

Jo

dtu and v = u+ W.

If we denote by /-t and v two new unknowns, such that we
have

a-x_ * ft-y_

;* I7T

and if we substitute for a, /?, dz, d@ their values

#4-2^7^, y + 2vjit 2d
we have this other form of the integral,

We could not multiply further these applications of our
formulae without diverging from our chief subject. The preceding
examples relate to physical phenomena, whose laws were un
known and difficult to discover; and we have chosen them because

422 THEORY OF HEAT. [CHAP. IX.

the integrals of these equations, which have hitherto been
fruitlessly sought for, have a remarkable analogy with those which
express the movement of heat.

413. We might also, in the investigation of the integrals,
consider first series developed according to powers of one variable,
and sum these series by means of the theorems expressed by the
equations (B), (BB). The following example of this analysis,
taken from the theory of heat itself, appeared to us to be
worthy of notice.

We have seen, Art. 399, that the general value of u derived
from the equation

dv d*v , N

dt=dj ...................... • ...... (a)>

developed in series, according to increasing powers of the variable
t, contains one arbitrary function only of x ; and that when de
veloped in series according to increasing powers of x, it contains
two completely arbitrary functions of t.

The first series is expressed thus :

v = t(*) + tJ2tW + ft^4>W + to-'--. ..... (T).

The integral denoted by (£), Art. 397, or

v = ^- \ dy. <j> (a) I dp e~pZ* cos (px — ^?a),

represents the sum of this series, and contains the single arbitrary
function <£ (as).

The value of v, developed according to powers of x, contains
two arbitrary functions f(t) and F(t), and is thus expressed :

There is therefore, independently of equation (/3), another
form of the integral which represents the sum of the last series,
and which contains two arbitrary functions, f(t) and F(f).

SECT. IV.] SECONDARY INTEGRAL OF LINEAR EQUATION. 423

It is required to discover this second integral of the proposed
equation, which cannot be more general than the preceding,
but which contains two arbitrary functions.

We can arrive at it by summing each of the two series which
enter into equation (X). Now it is evident that if we knew, in
the form of a function of x and t, the sum of the first series which
contains f(t), it would be necessary, after having multiplied it by
dx, to take the integral with respect to x, and to change f (t) into
F (t). We should thus find the second series. Further, it would
be enough to ascertain the sum of the odd terms which enter into
the first series : for, denoting this sum by /i, and the sum of all
the other terms by v, we have evidently

[* [* dp
= I ax \ dx -j- .
Jo Jo «

It remains then to find the value of p. Now the function
f(t) may be thus expressed, by means of the general equation (B\

It is easy to deduce from this the values of the functions

It is evident that differentiation is equivalent to writing in
the second member of equation (5), under the sign I dp, the
respective factors — p2, +p*, — p6, &c.

We have then, on writing once the common factor cos (pt—pz),

Thus the problem consists in finding the sum of the series
which enters into the second member, which presents no difficulty.
In fact, if y be the value of this series, we conclude

<?y 2 , p4^ p6^8 , i <2*y 5

^=-/+- • ors?=~py-

424 THEORY OF HEAT. [CHAP. IX.

Integrating this linear equation, and determining the arbitrary
constants, so that, when x is nothing, y may be 1, and

dij fry d?i/
tx> dx2' d?'

may be nothing, we find, as the sum of the series,

It would be useless to refer to the details of this investigation ;
it is sufficient to state the result, which gives, as the integral
sought,

v — - |cZa/(a) Idq q -jcos 2<?2 (t — a) (e^ + e~vx) cos qx
- sin 2^2 (t - a) (&* - e~qx] sin gx } + W. .....

The term W is the second part of the integral; it is formed by
integrating the first part with respect to x, from x = 0 to x = x,
and by changing / into F. Under this form the integral contains
two completely arbitrary functions f(t) and F (t). If, in the value
of v, we suppose x nothing, the term W becomes nothing by
hypothesis, and the first part u of the integral becomes f(t}. If

we make the same substitution x = 0 in the value of -r- it is

ax

evident that the first part -j- will become nothing, and that the

dx

dW
second, -j— , which differs only from the first by the function

F being substituted for ft will be reduced to F (t). Thus the
integral expressed by equation (00) satisfies all the conditions,
and represents the sum of the two series which form the second
member of the equation (X).

This is the form of the integral which it is necessary to select
in several problems of the theory of heat1; we see that it is very
different from that which is expressed by equation (/3), Art. 897.

1 See the article by Sir W. Thomson, "On the Linear Motion of Heat," Part II.
Art. 1. Camb. Math. Journal, Vol. III. pp. 206—8. [A. F.]

SECT. IV.] SERIES EXPRESSED BY DEFINITE INTEGRALS. 425

414. We may employ very different processes of investigation
to express, by definite integrals, the sums of series which repre
sent the integrals of differential equations. The form of these
expressions depends also on the limits of the definite integrals.
We will cite a single example of this investigation, recalling the
result of Art. 311. If in the equation which terminates that
Article we write x + 1 sin u under the sign of the function c£,
we have

1 l"du <j>(x + t sin u) - + (*);+ a </>" (x) + =Ai ^ (.r)

7T .. o 4 £ .*±

Denoting by v the sum of the series which forms the second
member, we see that, to make one of the factors 22, 42, 62, &c.
disappear in each term, we must differentiate once with respect
to t, multiply the result by t, and differentiate a second time with
respect to t. We conclude from this that v satisfies the partial
differential equation

d~v _l d^f dv\ d^v_(Fv Idv

dx* ~ 1 It ( t~dt) ' ™cU?~~d?+:tdt'

We have therefore, to express the integral of this equation,

1 [n
v = — I du (j> (x + 1 sin 11) + W.

The second part W of the integral contains a new arbitrary
function.

The form of this second part W of the integral differs very
much from that of the first, and may also be expressed by definite
integrals. The results, which are obtained by means of definite
integrals, vary according to the processes of investigation by which
they are derived, and according to the limits of the integrals.

415. It is necessary to examine carefully the nature of the
general propositions which serve to transform arbitrary functions :
for the use of these theorems is very extensive, and wre derive
from them directly the solution of several important physical
problems, which could be treated by no other method. The

426 THEORY OF HEAT. [CHAP. IX.

following proofs, which we gave in our first researches, are very
suitable to exhibit the truth of these propositions.

In the general equation

-i r+x> /•+<»

f(x) = - I cfaf (a) dp cos (py. -px)t

" « — oo JO

which is the same as equation (B), Art. 404, we may effect the in
tegration with respect to p, and we find

a-x

We ought then to give to p, in the last expression, an infinite
value; and, this being done, the second member will express the
value of f(&). We shall perceive the truth of this result by
means of the following construction. Examine first the definite

C m vi /y*

integral I dx — - , which we know to be equal to JTT, Art. 356.
Jo x

If we construct above the axis of x the curve whose ordinate is
sin x, and that whose ordinate is -, and then multiply the ordinate

M>

of the first curve by the corresponding ordinate of the second, we
may consider the product to be the ordinate of a third curve
whose form it is very easy to ascertain.

Its first ordinate at the origin is 1, and the succeeding ordinates
become alternately positive or negative; the curve cuts the axis
at the points where x = TT, 2?r, 3?r, &c., and it approaches nearer
and nearer to this axis.

A second branch of the curve, exactly like the first, is situated

r°° sin x

to the left of the axis of y. The integral I dx is the area

Jo af

included between the curve and the axis of x, and reckoned from
x — 0 up to a positive infinite value of x.

» 00 •

The definite integral / dx — — , in which p is supposed to be
Jo &

any positive number, has the same value as the preceding. In

fact, let px = z ; the proposed integral will become I dz — — , and,

Jo z
consequently, it is also equal to ^TT. This proposition is true,

SECT. IV.] AREAS REPRESENTING INTEGRALS. 427

whatever positive number p may be. If we suppose, for example,

1A ,, T ,. .sn ,

p = 10, the curve whose ordmate is - - has sinuosities very

x J

much closer and shorter than the sinuosities whose ordinate is
; but the whole area from x = 0 up to x = x is the same.

x

Suppose now that the number p becomes greater and greater,
and that it increases without limit, that is to say, becomes infinite.

The sinuosities of the curve whose ordinate is - — — are infinitely

ss

near. Their base is an infinitely small length equal to - . That
being so, if we compare the positive area which rests on one
of these intervals -- with the negative area which rests on the

following interval, and if we denote by JTthe finite and sufficiently
large abscissa which answers to the beginning of the first arc,
we see that the abscissa a?, which enters as a denominator into

the expression — of the ordinate, has no sensible variation in

the double interval — , which serves as the base of the two areas.

Consequently the integral is the same as if x were a constant
quantity. It follows that the sum of the two areas which succeed
each other is nothing.

The same is not the case wrhen the value of x is infinitely

small, since the interval — has in this case a finite ratio to the

P

r 01 •p T?*/1

value of x. We know from this that the integral / dx — , in

Jo *£

which we suppose^? to be an infinite number, is wholly formed out
of the sum of its first terms which correspond to extremely small
values of x. When the abscissa has a finite value X, the area
does not vary, since the parts which compose it destroy each other
two by two alternately. We express this result by writing

x

428 THEORY OF HEAT. [CHAP. IX.

The quantity «, which denotes the limit of the second integral,
has an infinitely small value ; and the value of the integral is the
same when the limit is co and when it is oo .

416. This assumed, take the equation

/, , N sin p (a. — x) . .

— - ^

, N 1 f +

*)»-]

Having laid down the. axis of the abscissae a, construct above
that axis the curve ff, whose ordinate is / (a). The form of
this curve is entirely arbitrary; it might have ordinates existing
only in one or several parts of its course, all the other ordinates
being nothing.

Construct also above the same axis of abscissae a curved line ss
whose ordinate is — — , z denoting the abscissa and p a very

great positive number. The centre of this curve, or the point
which corresponds to the greatest ordinate p, may be placed at the
origin 0 of the abscissae a, or at the end of any abscissa whatever.
We suppose this centre to be successively displaced, and to be
transferred to all points of the axis of or, towards the right, depart
ing from the point 0. Consider what occurs in a certain position
of the second curve, when the centre has arrived at the point x,
which terminates an abscissa x of the first curve.

The value of x being regarded as constant, and a being the
only variable, the ordinate of the second curve becomes

sin p (a — oc)
VL — X

If then we link together the two curves, for the purpose of
forming a third, that is to say, if we multiply each ordinate of the
second, and represent the product by an ordinate of a third curve
drawn above the axis of a, this product is

, , . sinp (a — a?)
** a — x

The whole area of the third curve, or the area included between
this curve and the axis of abscissae, may then be expressed by

7 /• / \ sin;? (a — x)
J a-x

SECT. IV.] EXAMINATION OF AX INTEGRAL. 429

Now the number p being infinitely great, the second curve has
all its sinuosities infinitely near ; we easily see that for all points
which are at a finite distance from the point x, the definite
integral, or the whole area of the third curve, is formed of equal
parts alternately positive or negative, which destroy each other two
by two. In fact, for one of these points situated at a certain dis
tance from the point #, the value of /(a) varies infinitely little

when we increase the distance by a quantity less than — . The
same is the case with the denominator a — x, which measures that

distance. The area which corresponds to the interval — is there-

P
fore the same as if the quantities /(a) and a — a; were not variables.

Consequently it is nothing when a — x is a finite magnitude.
Hence the definite integral may be taken between limits as near
as we please, and it gives, between those limits, the same result as
between infinite limits. The whole problem is reduced then to
taking the integral between points infinitely near, one to the left,
the other to the right of that where a — x is nothing, that is to say
from OL = X — co to a = x+ co, denoting by co a quantity infinitely
small. In this interval the function /(a) does not vary, it is
equal to/ (a?), and may be placed outside the symbol of integra
tion. Hence the value of the expression is the product of f(jc) by

[

J

a — x
taken between the limits a — x = — co, and a — x = co.

Now this integral is equal to TT, as we have seen in the pre
ceding article ; hence the definite integral is equal to irf(x)t whence
we obtain the equation

*/ \ 1 r* j s / \ ^ sin p (a. — x} , .

/<*) = 5z / <**/<«) - "irjr '<**•*)

— O

-i») ...... (B).

J -co * ""CO

417. The preceding proof supposes that notion of infinite
quantities which has always been admitted by geometers. It
would be easy to offer the same proof under another form, examin
ing the changes which result from the continual increase of the

430 THEORY OF HEAT. [CHAP. IX.

factory under the symbol sin/> (OL — X). These considerations are
too well known to make it necessary to recall them.

Above all, it must be remarked that the function /(a?), to which
this proof applies, is entirely arbitrary, and not subject to a con
tinuous law. We might therefore imagine that the enquiry is
concerning a function such that the ordinate which represents it
has no existing value except when the abscissa is included between
two given limits a and b, all the other ordinates being supposed
nothing ; so that the curve has no form or trace except above the
interval from x = a to x = b, and coincides with the axis of a in
all other parts of its course.

The same proof shews that we are not considering here infinite
values of x, but definite actual values. We might also examine on
the same principles the cases in which the function f(x) becomes
infinite, for singular values of x included between the given limits;
but these have no relation to the chief object which we have in
view, which is to introduce into the integrals arbitrary functions ;
it is impossible that any problem in nature should lead to the
supposition that the function f(x) becomes infinite, when we
give to a; a singular value included between given limits.

In general the function f(x) represents a succession of values
or ordinates each of which is arbitrary. An infinity of values being
given to the abscissa x, there are an equal number of ordinates
/ (x). All have actual numerical values, either positive or negative
or nul.

We do not suppose these ordinates to be subject to a common
law; they succeed each other in any manner whatever, and each of
them is given as if it were a single quantity.

It may follow from the very nature of the problem, and from
the analysis which is applicable to it, that the passage from one
ordinate to the following is effected in a continuous manner. But
special conditions are then concerned, and the general equation (B),
considered by itself, is independent of these conditions. It is
rigorously applicable to discontinuous functions.

Suppose now that the function f(x) coincides with a certain
analytical expression, such as sina«, e~x\ or $ (x), when we give to
x a value included between the two limits a and b, and that all

SECT. IV.] FUNCTIONS COINCIDING BETWEEN LIMITS. 431

the values of f(x] are nothing when x is not included between a
and 6; the limits of integration with respect to a, in the preceding
equation (B\ become then a = a, a = 6; since the result is the same
as for the limits a = — oc , a = oo , every value of </> (a) being nothing
by hypothesis, when a is not included between a and b. We have
then the equation

The second member of this equation (B') is a function of the
variable x\ for the two integrations make the variables a. andp dis
appear, and x only remains with the constants a and b. Now the
function equivalent to the second member is such, that on substitut
ing for x any value included between a and b, we find the same
result as on substituting this value of x in <f> (x) ; and we find a nul
result if, in the second member, we substitute for x any value not
included between a and b. If then, keeping all the other quantities
which form the second member, we replaced the limits a and b
by nearer limits a and &', each of which is included between a and
6, we should change the function of x which is equal to the second
member, and the effect of the change would be such that the
second member would become nothing whenever we gave to # a
value not included between d and 6'; and, if the value of x were
included between a and 6', we should have the same result as
on substituting this value of x in <j>(x).

We can therefore vary at will the limits of the integral in the
second member of equation (B"). This equation exists always for
values of x included between any limits a and b, which we may
have chosen; and, if we assign any other value to x, the second
member becomes nothing. Let us represent <t>(x) by the variable
ordinate of a curve of which x is the abscissa ; the second member,
whose value is /(a?), will represent the variable ordinate of a second
curve whose form will depend on the limits a and b. If these
limits are — oc and + 20 , the two curves, one of which has <j>(x) for
ordinate, and the other f(x], coincide exactly through the whole
extent of their course. But, if we give other values a and b to these
limits, the two curves coincide exactly through every part of their
course which corresponds to the interval from x = a to x = b. To
right and left of this interval, the second curve coincides precisely

432 THEORY OF HEAT. [CHAP. IX.

at every point with the axis of x. This result is very remarkable,
and determines the true sense of the proposition expressed by
equation (B).

418. The theorem expressed by equation (II) Art. 234 must
be considered under the same point of view. This equation
serves to develope an arbitrary function / (x) in a series of sines or
cosines of multiple arcs. The function f(x) denotes a function
completely arbitrary, that is to say a succession of given values,
subject or not to a common law, and answering to all the values of
x included between 0 and any magnitude X.

The value of this function is expressed by the following
equation,

«*?y(*-lO (A).

The integral, with respect to a, must be taken between the
limits a = a, and a = 6 ; each of these limits a and I is any quantity
whatever included between 0 and X. The sign 2 affects the
integer number it and indicates that we must give to i every
integer value negative or positive, namely,

...-5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5,...

and must take the sum of the terms arranged under the sign 2.
After these integrations the second member becomes a function of
the variable x only, and of the constants a and b. The general
proposition consists in this : 1st, that the value of the second
member, which would be found on substituting for x a quantity
included between a and &, is equal to that which would be obtained
on substituting the same quantity for x in the function /(a?); 2nd,
every other value of x included between 0 and X, but not included
between a and b, being substituted in the second member, gives a
mil result.

Thus there is no function f(x), or part of a function, which
cannot be expressed by a trigonometric series.

The value of the second member is periodic, and the interval
of the period is X, that is to say, the value of the second member
does not change when x + X is written instead of x. All its
values in succession are renewed at intervals X.

SECT. IV.] TRANSFORMATION OF FUNCTIONS. 433

The trigonometrical series equal to the second member is
convergent; the meaning of this statement is, that if we give to
the variable x any value whatever, the sum of the terms of the
series approaches more and more, and infinitely near to, a definite
limit. This limit is 0, if we have substituted for x a quantity
included between 0 and X, but not included between a and ft;
but if the quantity substituted for x is included between a and b,
the limit of the series has the same value as f(x). The last
function is subject to no condition, and the line whose ordinate it
represents may have any form; for example, that of a contour
formed of a series of straight lines and curved lines. We see by
this that the limits a and b, the w^hole interval X, and the nature
of the function being arbitrary, the proposition has a very exten
sive signification ; and, as it not only expresses an analytical
property, but leads also to the solution of several important
problems in nature, it wras necessary to consider it under different
points of view, and to indicate its chief applications. We have
given several proofs of this theorem in the course of this work.
That which we shall refer to in one of the following Articles
(Art. 424) has the advantage of being applicable also to non-
periodic functions.

If we suppose the interval X infinite, the terms of the series
become differential quantities ; the sum indicated by the sign 2
becomes a definite integral, as was seen in Arts. 353 and 355, and
equation (A) is transformed into equation (B). Thus the latter
equation (B) is contained in the former, and belongs to the case
in which the interval X is infinite: the limits a and b are then
evidently entirely arbitrary constants.

419. The theorem expressed by equation (B) presents also
divers analytical applications, which we could not unfold without
quitting the object of this work; but we will enunciate the
principle from which these applications are derived.

We see that, in the second member of the equation

the function f(x) is so transformed, that the symbol of the

function / affects no longer the variable &, but an auxiliary

F. H. 28

434 THEORY OF HEAT. [CHAP, IX.

variable a. The variable x is only affected by the symbol cosine.
It follows from this, that in order to differentiate the function / (x)
with respect to x, as many times as we wish, it is sufficient to
differentiate the second member with respect to a under the
symbol cosine. We then have, denoting by i any integer number
whatever,

We take the upper sign when i is even, and the lower sign
when i is odd. Following the same rule relative to the choice

of sign

We can also integrate the second member of equation (Z?)
several times in succession, with respect to x\ it is sufficient to
write in front of the symbol sine or cosine a negative power
of p.

The same remark applies to finite differences and to summa
tions denoted by the sign 2, and in general to analytical operations
which may be effected upon trigonometrical quantities. The chief
characteristic of the theorem in question, is to transfer the general
sign of the function to an auxiliary variable, and to place the
variable x under the trigonometrical sign. The function f(x)
acquires in a manner, by this transformation, all the properties of
trigonometrical quantities ; differentiations, integrations, and sum
mations of series thus apply to functions in general in the same
manner as to exponential trigonometrical functions. For which
reason the use of this proposition gives directly the integrals
of partial differential equations with constant coefficients. In
fact, it is evident that we could satisfy these equations by par
ticular exponential values ; and since the theorems of which we
are speaking give to the general and arbitrary functions the
character of exponential quantities, they lead easily to the expres
sion of the complete integrals.

The same transformation gives also, as we have seen in
Art. 413, an easy means of summing infinite series, when these
series contain successive differentials, or successive integrals of the

SECT. IV.] REAL AND UNREAL PARTS OF A FUNCTION. 43.")

same function ; for the summation of the series is reduced, by
what precedes, to that of a series of algebraic terms.

420. We may also employ the theorem in question for the
purpose of substituting under the general form of the function a
binomial formed of a real part and an imaginary part. This
analytical problem occurs at the beginning of the calculus of
partial differential equations ; and we point it out here since it
has a direct relation to our chief object.

If in the function f(x) we write \L + v — 1 instead of #, the
result consists of two parts (b+J—lty. The problem is to
determine each of these functions </> and ty in terms of //. and v.
We shall readily arrive at the result if we replace f(x) by the
expression

for the problem is then reduced to the substitution of /A + v — I
instead of x under the symbol cosine, and to the calculation of the
real term and the coefficient of — 1. We thus have

=/(/* + v J~l) = ~jdz (*) fdp cos [p (p - a) +pv
4~ pa/(a) I

cos ~* epv + e~pv

l sn -

hence $ = — |d«/(a) [dp cos (pp -pz)

Thus all the functions f(x) which can be imagined, even those
which are not subject to any law of continuity, are reduced to the
form M -f- Nj— 1, when we replace the variable x in them by the
binomial yu,+ v*J- 1.

28—2

436 THEOEY OF HEAT. [CHAP. IX.

421. To give an example of the use of the last two formulae,
let us consider the equation -^ + -,— ^ = 0, which relates to the

uniform movement of heat in a rectangular plate. The general
integral of this equation evidently contains two arbitrary func
tions. Suppose then that we know in terms of x the value of v
when y = 0, and that we also know, as another function of x, the

value of -7- when y = 0, we can deduce the required integral from
that of the equation

which has long been known; but we find imaginary quantities
under the functional signs : the integral is

v = </> (x + y^l) + <£ (x - 2/7=3) + W.

The second part W of the integral is derived from the first by
integrating with respect to y, and changing <f> into ^r.

It remains then to transform the quantities $(x + y J— 1) and
$ (# ~~ yj~ i)> m order to separate the real parts from the ima
ginary parts. Following the process of the preceding Article we
find for the first part u of the integral,

1 /•+» r+30

u = ^- I da/(a) I dp cos (px -pa) (e

» —00 ^ —GO

and consequently

W= & F(a) cos (p« - iw) (e- - e-)-

The complete integral of the proposed equation expressed in
real terms is therefore v = u + W ; and we perceive in fact,
•1st, that it satisfies the differential equation ; 2nd, that on making
y = 0 in it, it gives v =f(x) ; 3rd, that on making y — 0 in the

function -7- , the result is F(x).

SECT. IV.] DIFFERENTIATION OF FUNCTIONS. 437

422. We may also remark that we can deduce from equation
(B) a very simple expression of the differential coefficient of the

dl [*

ith order, -T-j/OOi or of the integral I dxlf(x).

The expression required is a certain function of x and of the
index i. It is required to ascertain this function under a form
such that the number i may not enter it as an index, but as a
quantity, in order to include, in the same formula, every case in
which we assign to i any positive or negative value. To obtain it
we shall remark that the expression

cos

^7^ . ITT

or ' cos r cos -^ — sin r sin -=- ,

4 A

becomes successively

- sin r, - cos r, + sin r, + cos r, — sin r, &c.,

if the respective values of i are 1, 2, 3, 4, 5, &c. The same results
recur in the same order, when we increase the value of i. In the
second member of the equation

cos x ~

we must now write the factor p* before the symbol cosine, and
add under this symbol the term -f- i- . We shall thus have

The number i, which enters into the second member, may be
any positive or negative integer. We shall not press these applica
tions to general analysis ; it is sufficient to have shewn the use of
our theorems by different examples. The equations of the fourth
order, (d\ Art, 405, and (e), Art. 411, belong as we have said to
dynamical problems. The integrals of these equations were not
yet known when we gave them in a Memoir on the Vibrations of

438 THEOKY OF HEAT. [CHAP. IX.

Elastic Surfaces, read at a sitting of the Academy of Sciences1,
Gth June, 1816 (Art. VI. §§ 10 and 11, and Art. vii. §§ 13 and 14).
They consist in the two formulae S and 8', Art. 40G, and in the two
integrals expressed, one by the first equation of Art. 412, the other
by the last equation of the same Article. We then gave several
other proofs of the same results. This memoir contained also the
integral of equation (c), Art. 409, under the form referred to in
that Article. "With regard to the integral (/3/3) of equation (a),
Art. 413, it is here published for the first time.

423. The propositions expressed by equations (A) and (B'),
Arts. 418 and 417, may be considered under a more general point
of view. The construction indicated in Arts. 415 and 41 G applies

Sill f ?)j£ ^-^ 77 71)

not only to the trigonometrical function - — — — - ; but suits

oc — oc

all other functions, and supposes only that when the number p
becomes infinite, we find the value of the integral with respect to
a, by taking this integral between extremely near limits. Now
this condition belongs not only to trigonometrical functions, but is
applicable to an infinity of other functions. We thus arrive at
the expression of an arbitrary function f(x) under different very
remarkable forms ; but we make no use of these transformations
in the special investigations which occupy us.

With respect to the proposition expressed by equation (A),
Art. 418, it is equally easy to make its truth evident by con
structions, and this was the theorem for which we employed them
at first. It will be sufficient to indicate the course of the proof.

1 The date is inaccurate. The memoir was read on June 8th, 1818, as appears
from an abstract of it given in the Bulletin dcs Sciences par la Societe Philomatique,
September 1818, pp. 129 — 136, entitled, Note relative mix vibrations des surfaces
elastiques et au mouvement des ondes, par M. Fourier. The reading of the memoir
further appears from the Analyse des travaux de V Academic des Sciences pendant
Vannee 1818, p. xiv, and its not having been published except in abstract, from a
remark of Poissoii at pp. 150 — 1 of his memoir Sur les Equations aux differences
partielles, printed in the Memoires de VAcademie des Sciences, Tome in. (year 1818),
Paris, 1820. The title, Memoire sur les vibrations des surfaces glastiques, par
M. Fourier, is given in the Analyse, p. xiv. The object, "to integrate several
partial differential equations and to deduce from the integrals the knowledge of the
physical phenomena to which these equations refer," is stated in the Bulletin,
p. ISO. LA. I'1.]

SECT. IV.] EXAMINATION OF AN INTEGRAL. 439

In equation (A), namely,

we can replace the sum of the terms arranged under the
sign 2 by its value, which is derived from known theorems.
We have seen different examples of this calculation previously,
Section III., Chap. in. It gives as the result if we suppose,
in order to simplify the expression, 2?r = X, and denote a-#
by r,

_-+.; . . . sin r

2j cos ir = cos ?r+ sin ir - -.
-j J versmr

We must then multiply the second member of this equation
by cZx/(a), suppose the number j infinite, and integrate from
a = - TT to a = + TT. The curved line, whose abscissa is a and
ordinate cos^V, being conjoined with the line whose abscissa is
a. and ordinate /(a), that is to say, when the corresponding
ordinates are multiplied together, it is evident that the area of
the curve produced, taken between any limits, becomes nothing
when the number j increases without limit. Thus the first term
cosjr gives a nul result.

The same would be the case with the term sinjr, if it were
not multiplied by the factor - ^ — ; but on comparing the
three curves which have a common abscissa a, and as ordinates

sm r
versin r ' '

sin ?

, we see clearly that the integral

•

c/a/(a) sinjV

versiii r

has no actual values except for certain intervals infinitely small,

namely, when the ordinate - • becomes infinite. This will

versin ?*

take place if r or a — x is nothing ; and in the interval in which
a differs infinitely little from x, the value of /(a) coincides with
f(x). Hence the integral becomes

J0

r sin Jr > or 4/(.r) j ~ sin jr,

44-0 THEORY OF HEAT. [CHAP. IX.

which is equal to 2irf(x)t Arts. 415 and 350. Whence we con
clude the previous equation (A).

When the variable x is exactly equal to — TT or + TT, the con
struction shews what is the value of the second member of the
equation (A), [|/(-7r) or ^/(TT)].

If the limits of integrations are not - TT and + TT, but other I
numbers a and b, each of which is included between — TT and
+ TT, we see by the same figure what the values of x are, for which
the second member of equation (A) is nothing.

If we imagine that between the limits of integration certain
values of /(a) become infinite, the construction indicates in what
sense the general proposition must be understood. But we do
not here consider cases of this kind, since they do not belong
to physical problems.

If instead of restricting the limits — TT and + TT, we give
greater extent to the integral, selecting more distant limits a
and b', we know from the same figure that the second member
of equation (A) is formed of several terms and makes the result
of integration finite, whatever the function /(#) may be.

We find similar results if we write 2?r —y— instead of r, the

limits of integration being — X and + X.

It must now be considered that the results at which we
have arrived would also hold for an infinity of different functions
of sin jr. It is sufficient for these functions to receive values
alternately positive and negative, so that the area may become
nothing, when j increases without limit. We may also vary

the factor —. — -, as well as the limits of integration, and we
versm r

may suppose the interval to become infinite. Expressions of
this kind are very general, and susceptible of very different forms.
We cannot delay over these developments, but it was necessary
to exhibit the employment of geometrical constructions ; for
they solve without any doubt questions which may arise on the
extreme values, and on singular values; they would not have
served to discover these theorems, but they prove them and guide
all their applications.

SECT. IV.] DEVELOPMENT IX SERIES OF FUNCTIONS. 441

424. We have yet to regard the same propositions under
another aspect. If we compare with each other the solutions
relative to the varied movement of heat in a ring, a sphere, a
rectangular prism, a cylinder, we see that we had to develope
an arbitrary function f(x) in a series of terms, such as

«i</> OvO + <v£ (/v*0 + °3<!> (/vO + &c-

The function (£, which in the second member of equation
(A) is a cosine or a sine, is replaced here by a function which
may be very different from a sine. The numbers filt //,2, //,3, &c.
instead of being integers, are given by a transcendental equation,
all of whose roots infinite in number are real

The problem consisted in finding the values of the coefficients
a\> av as - - - ai I they nave been arrived at by means of definite
integrations which make all the unknowns disappear, except one.
We proceed to examine specially the nature of this process, and
the exact consequences which flow from it.

In order to give to this examination a more definite object,
we will take as example one of the most important problems,
namely, that of the varied movement of heat in a solid sphere.
\Ve have seen, Art. 290, that, in order to satisfy the initial dis
tribution of the heat, we must determine the coefficients al} aa,
r/-s ... ai? in the equation

ocF(x) = at sin (^x) + a2 sin (JLL^X) -4- a3 sin (p3x) + &c ....... (e).

The function F(x) is entirely arbitrary ; it denotes the value
v of the given initial temperature of the spherical shell whose
radius is x. The numbers /^, /z-a ... p. are the roots /^, of the
transcendental equation

X is the radius of the whole sphere; h is a known numerical co
efficient having any positive value. We have rigorously proved in
our earlier researches, that all the values of fju or the roots of the
equation (/) are real1. This demonstration is derived from the

1 The Mfrnoircs de V Academic des Sciences, Toine x, Paris 1831, pp. 119 — 146,
contain Rcmarqiifs fjcncralc* sur V application des principes dc Vanalyse algebriquc

442 THEORY OF HEAT. [CHAP. IX.

general theory of equations, and requires only that we should
suppose known the form of the imaginary roots which every equa
tion may have. We have not referred to it in this work, since its
place is supplied by constructions which make the proposition more
evident. Moreover, we have treated a similar problem analytically,
in determining the varied movement of heat in a cylindrical body
(Art. 308). This arranged, the problem consists in discovering
numerical values for alt #2, ag,...af, &c., such that the second
member of equation (e) necessarily becomes equal to xF(x), when
we substitute in it for x any value included between 0 and the
whole length X.

To find the coefficient «., we have multiplied equation (e) by
dx sin fita;, and then integrated between the limits x — 0, x = X,
and we have proved (Art. 291) that the integral

rX

I dx sin figc sin ^x

Jo

has a null value whenever the indices i and j are not the same;
that is to say when the numbers pi and /*, are two different roots
of the equation (/). It follows from this, that the definite inte
gration making all the terms of the second member disappear,
except that which contains ait we have to determine this coefficient,
the equation

x ix

dx \x F (x\ sin pp] = a.l dx sin pp sin pp.

o Jo

Substituting this value of the coefficient at in equation (e), we
derive from it the identical equation (e),

x
dot. a,F(a) s

r
I

Jo

r
I

d@ sin a & sin a B

Jo

aux equations transcendantes , by Fourier. The author shews that the imaginary
roots of sec x=Q do not satisfy the equation tance=0, since for them, tan# = JN/ - 1.
The equation tan x = 0 is satisfied only by the roots of sin x — 0, which are all real.
It may be shewn also that the imaginary roots of sec # = 0 do not satisfy the equation
x-mtsinx-Q, where m is less than 1, but this equation is satisfied only by the
roots of the equation f(x) = x cos x - m s'mx = 0, which are all real. For if
fr+1(x), fr(x], fr-i(x), are three successive differential coefficients of f(x), the values
of x which make fr («)=0, make the signs of /r+1 (x) and /r-1 (x) different. Hence
by Fourier's Theorem relative to the number of changes of sign of f(x) and its
successive derivatives, /(.r) can have no imaginary roots. [A. F.j

SECT. IV.] WHAT TERMS MUST BE INCLUDED. 443

In the second member we must give to i all its values, that is to
say we must successively substitute for ^, all the roots p, of the
equation (/). The integral must be taken for a from a = 0 to
a = X, which makes the unknown a disappear. The same is the
case with /3, which enters into the denominator in such a manner
that the term sin p.x is multiplied by a coefficient a. whose value
depends only on X and on the index i. The symbol S denotes
that after having given to i its different values, we must write
down the sum of all the terms.

The integration then offers a very simple means of determining
the coefficients directly; but we must examine attentively the
origin of this process, which gives rise to the following remarks.

1st. If in equation (e) we had omitted to write down part of
the terms, for example, all those in which the index is an even
number, we should still find, on multiplying the equation by
dx sin fj,.x, and integrating from x = 0 to x = X, the same value of
an which has been already determined, and we should thus form
an equation which would not be true ; for it would contain only
part of the terms of the general equation, namely, those whose
index is odd.

2nd. The complete equation (e) which we obtain, after having
determined the coefficients, and which does not differ from the
equation referred to (Art. 291) in which we might make £=0 and
v =/(#), is such that if we give to x any value included between
0 and X, the two members are necessarily equal; but we cannot
conclude, as we have remarked, that this equality would hold, if
choosing for the first member xF (x) a function subject to a con
tinuous law, such as sin x or cos x, we were to give to x a value
not included between 0 and X. In general the resulting equation
(e) ought to be applied to values of x, included between 0 and ^Y.
Now the process which determines the coefficient at does not
explain why all the roots ^ must enter into equation (e), nor
why this equation refers solely to values of a:, included between 0
and X.

To answer these questions clearly, it is sufficient to revert to
the principles which serve as the foundation of our analysis.

We divide the interval X into an infinite number n of parts

444 THEORY OF HEAT. [CHAP. IX.

equal to dx, so that we have ndx = X, and writing f (x) instead of
xF(x),wQ denote by /^/^jf. .../;.../„, the values of /(#), which
correspond to the values dx, 2dx, Sdx, . . . idx . . . ndx, assigned to
x ; we make up the general equation (e) out of a number n of
terms; so that n unknown coefficients enter into it, av a2, «3, ...
^...a^ This arranged, the equation (e) represents n equations
of the first degree, which we should form by substituting succes
sively for x, its n values dx, 2dx, 3dx,...ndx. This system of n
equations contains yj in the first equation, /2 in the second, /3 in
the third, fn in the nih. To determine the first coefficient alt we
multiply the first equation by a-lt the second by cr2, the third by
<73, and so on, and add together the equations thus multiplied.
The factors <71} cr2, o-g, ...o-tt must be determined by the condition,
that the sum of all the terms of the second members which contain
aa must be nothing, and that the same shall be the case with the
following coefficients aa, c&4, ...an. All the equations being then
added, the coefficient a^ enters only into the result, and we have
an equation for determining this coefficient. We then multiply
all the equations anew by other factors pl, p2, p3,...pn respectively,
and determine these factors so that on adding the n equations, all
the coefficients may be eliminated, except a2. We have then an
equation to determine a2. Similar operations are continued, and
choosing always new factors, we successively determine all the
unknown coefficients. Now it is evident that this process of elimi
nation is exactly that which results from integration between the
limits 0 and X. The series <rl, cr2, <r3,...<rn of the first factors is
dx sin (fijdx), dx sin (p^dx), dx sin (pfidx) ...dx sin (^ndx). In
general the series of factors which serves to eliminate all the co
efficients except ait is dx sin (^dx), dx sin (>. 2dr), dx sin (^ 3dx) . . .
dx sin (pjridx) ; it is represented by the general term dx sin (^x),
in which we give successively to x all the values

dx, 2f&, %dx, . . . ndx.

We see by this that the process which serves to determine these
coefficients, differs in no respect from the ordinary process of elimi
nation in equations of the first degree. The number n of equations
is equal to that of the unknown quantities alf «2, aa...an, and is
the same as the number of given quantities /,,/,,/,... /^ The
values found for the coefficients are those which must exist in

SECT. IV.] CONDITIONS OF DEVELOPMENT. 445

order that the n equations may hold good together, that is to say
in order that equation (e) may be true when we give to x one of
these n values included between 0 and X ; and since the number
n is infinite, it follows that the first member f (x) necessarily coin
cides with the second, when the value of x substituted in each
is included between 0 and X.

The foregoing proof applies not only to developments of the
form

a sin jLs + « sin x + a sin z# + . . . + a sin ,

it applies to all the functions <£ (frx) which might be substituted
for sin (/v&), maintaining the chief condition, namely, that the

integral f dx $ (pp) $ (/A/C) has a nul value when i and j are

Jo
different numbers.

If it be proposed to develope/(#) under the form

a, cos x a, cos 2j? a.cosix „

» ' +7 ' O +.-.+ / • + &C.,

b sm x 6 sm 2x b cos ix

the quantities plf /z2, ^3...^, &c. will be integers, and the con
dition

I e£c cos f2wt «.] sin f 2?rj -^J = 0,

always holding when the indices i and j are different numbers, we
obtain, by determining the coefficients at, biy the general equation
(II), page 206, which does not differ from equation (A) Art. 418.

425. If in the second member of equation (e) we omitted one
or more terms which correspond to one or more roots /^ of the
equation (/), equation (e) would not in general be true. To
prove this, let us suppose a term containing /^ and a, not to be
written in the second member of equation (e), we might multiply
the n equations respectively by the factors

dxsm(fijda:)9 dxsmfajZdx), dx sin (//_. 3dar) . . . dx sin fondx) ;

and adding them, the sum of all the terms of the second members
would be nothing, so that not one of the unknown coefficients
would remain. The result, formed of the sum of the first members,

446 THEORY OF HEAT. {CHAP. IX.

that is to say the sum of the values /, /2, /3 .../„, multiplied
respectively by the factors

dx sin (fjLjdx), dx sin (fjifidx], dx sin (pfidx) . . . dx sin (^ndx),

would be reduced to zero. This relation would then necessarily
exist between the given quantities/, , /2, /3 •••/„; and they could not
be considered entirely arbitrary, contrary to hypothesis. If these
quantities /, f2,fs---fn have any values whatever, the relation in
question cannot exist, and we cannot satisfy the proposed con
ditions by omitting one or more terms, such as a-3 sin (fijX) in
equation (e).

Hence the function f(x) remaining undetermined, that is to
say, representing the system of an infinite number of arbitrary
constants which correspond to the values of x included between 0
and X, it is necessary to introduce into the second member of
equation (e) all the terms such as a. sinter), which satisfy the
condition

x

dx sin /Aft sin fifx — 0,
o

the indices i and j being different; but if it happen that the
function /(*•) is such that the n magnitudes /,/2,/3 •-/» are
connected by a relation expressed by the equation

-x

dx sin fj,jxf(x) = 0,
o

it is evident that the term c^sin/*^ might be omitted in the equa
tion (e).

Thus there are several classes of functions / (x) whose develop
ment, represented by the second member of the equation (e), does
not contain certain terms corresponding to some of the roots JJL.
There are for example cases in which we omit all the terms
whose index is even; and we have seen different examples of this
in the course of this work. But this would not hold, if the func
tion /(a?) had all the generality possible. In all these cases, we
ought to suppose the second member of equation (e) to be com
plete, and the investigation shews what terms ought to be omitted,
since their coefficients become nothing.

SECT. IV.] SYSTEM OF QUANTITIES REPRESENTED. 447

426. We see clearly by this examination that the function /(.r)
represents, in our analysis, the system of a number n of separate
quantities, corresponding to n values of x included between 0 and
X, and that these n quantities have values actual, and consequently
not infinite, chosen at will. All might be nothing, except one,
whose value would be given.

It might happen that the series of the n values fltf2,fs .../„
was expressed by a function subject to a continuous law. such as
x or x3, sin#, or cos a-, or in general <j> (x) ; the curve line 0(70,
whose ordinates represent the values corresponding to the abscissa
x, and which is situated above the interval from x = 0 to x = X,
coincides then in this interval with the curve whose ordinate is
</> (x), and the coefficients alt a8, a3 ... an of equation (e) determined
by the preceding rule always satisfy the condition, that any value
of x included between 0 and X, gives the same result when substi
tuted in <p (x)-, and in the second member of equation (e).

F(x) represents the initial temperature of the spherical shell
whose radius is x. "We might suppose, for example, F(x) = bx,
that is to say, that the initial heat increases proportionally to the
distance, from the centre, where it is nothing, to the surface
where it is bX. In this case xF(x) or f(x) is equal to bx2; and
applying to this function the rule which determines the coeffi
cients, bx* would be developed in a series of terms, such as

al sin fax) + a2 sin fax) + az sin fax) + ... + an sin fax).

Now each term sinQ^oj), when developed according to powers
of x, contains only powers of odd order, and the function bx* is
a power of even order. It is very remarkable that this function
bxz, denoting a series of values given for the interval from 0
to X, can be developed in a series of terms, such as at sin fax).

We have already proved the rigorous exactness of these
results, which had not yet been presented in analysis, and we
have shewn the true meaning of the propositions which express
them. We have seen, for example, in Article 223, that the
function cos# is developed in a series of sines of multiple arcs,
so that in the equation which gives this development, the first
member contains only even powers of the variable, and the second
contains only odd powers. Reciprocally, the function sin x, into

r
I

448 THEORY OF HEAT. [CHAP. IX.

which only odd powers enter, is resolved, Art. 225, into a series
of cosines which contain only even powers.

In the actual problem relative to the sphere, the value of
xF(x) is developed by means of equation (e). We must then,
as we see in Art. 290, write in each term the exponential factor,
which contains t, and we have to express the temperature v,
which is a function of x and t, the equation

x
dxsin (fai) aF(ca)

.. ...... (E).

£ sin (/i40) sin fo/3)

The general solution which gives this equation (E} is wholly
independent of the nature of the function F(x) since this function
represents here only an infinite multitude of arbitrary constants,
which correspond to as many values of x included between 0
and X.

If we supposed the primitive heat to be contained in a part
only of the solid sphere, for example, from x = 0 to x = $X,
and that the initial temperatures of the upper layers were nothing,
it would be sufficient to take the integral

sin (^a )/(«),

between the limits x = 0 and x = ^X.

In general, the solution expressed by equation (E) suits all
cases, and the form of the development does not vary according to
the nature of the function.

Suppose now that having written sin x instead of F(x) we have
determined by integration the coefficients at) and that we have
formed the equation

x sin x = at sin JJL^X + «2 sin JJLZ% + a3 sin JJL^X -f &c.

It is certain that on giving to x any value whatever included
between 0 and X, the second member of this equation becomes
equal to a; since; this is a necessary consequence of our process.
But it nowise follows that on giving to a; a value not included
between 0 and X, the same equality would exist. We see the
contrary very distinctly in the examples which we have cited, and,

SECT. IV.] SINGLE LAYER INITIALLY HEATED. 449

particular cases excepted, we may say that a function subject to a
continuous law, which forms the first member of equations of this
kind, does not coincide with the function expressed by the second
member, except for values of x included between 0 and X.

Properly speaking, equation (e) is an identity, which exists
for all values which may be assigned to the variable x\ each
member of this equation representing a certain analytical function
which coincides with a known function f(x) if we give to the
variable x values included between 0 and A7". With respect to the
existence of functions, wThich coincide for all values of the variable
included between certain limits and differ for other values, it is
proved by all that precedes, and considerations of this kind are a
necessary element of the theory of partial differential equations.

Moreover, it is evident that equations (e) and (E) apply not
only to the solid sphere whose radius is X, but represent, one the
initial state, the other the variable state of an infinitely extended
solid, of which the spherical body forms part ; and when in these
equations we give to the variable x values greater than X,
they refer to the parts of the infinite solid which envelops the
sphere.

This remark applies also to all dynamical problems which are
solved by means of partial differential equations.

427. To apply the solution given by equation (E) to the case
in which a single spherical layer has been originally heated, all
the other layers having nul initial temperature, it is sufficient to

take the integral \dj. sin (/^a) aF (a) between two very near limits,

a = r, and a = r + u, r being the radius of the inner surface of the
heated layer, and u the thickness of this layer.

We can also consider separately the resulting effect of the
initial heating of another layer included between the limits r + u
and r + 2u ; and if we add the variable temperature due to this
second cause, to the temperature which we found when the first
layer alone was heated, the sum of the two temperatures is that
which would arise, if the two layers were heated at the same time.
In order to take account of the two joint causes, it is sufficient to
F. H. 29

450 THEORY OF HEAT. [CHAP. IX.

take the integral Ida sin (/i4ot) aF(a) between the limits a — r and

a = r + 2w. More generally, equation (E) being capable of being
put under the form

fxj -vi \ •
v = I ay. . ctr (a) sin /^a

0

sm W e

x \ d/3 si
Jo

sn uj sn

we see that the whole effect of the heating of different layers is
the sum of the partial effects, which would be determined separately,
by supposing each of the layers to have been alone heated. The
same consequence extends to all other problems of the theory of
heat ; it is derived from the very nature of equations, and the form
of the integrals makes it evident. We see that the heat con
tained in each element of a solid body produces its distinct effect,
as if that element had alone been heated, all the others having
nul initial temperature. These separate states are in a manner
superposed, and unite to form the general system of temperatures.

For this reason the form of the function which represents the
initial state must be regarded as entirely arbitrary. The definite
integral which enters into the expression of the variable tempera
ture, having the same limits as the heated solid, shows expressly
that we unite all the partial effects due to the initial heating of
each element.

428. Here we shall terminate this section, which is devoted
almost entirely to analysis. The integrals which we have obtained
are not only general expressions which satisfy the differential equa
tions ; they represent in the most distinct manner the natural effect
which is the object of the problem. This is the chief condition which
we have always had in view, and without which the results of in
vestigation would appear to us to be only useless transformations.
When this condition is fulfilled, the integral is, properly speaking,
the equation of the phenomenon; it expresses clearly the character
and progress of it, in the same manner as the finite equation of a
line or curved surface makes known all the properties of those
forms. To exhibit the solutions, we do not consider one form only
of the integral ; we seek to obtain directly that which is suitable
to the problem. Thus it is that the integral which expresses the

SECT. IV.] ELEMENTS OF THE METHOD PURSUED. 451

movement of heat in a sphere of given radius, is very different
from that which expresses the movement in a cylindrical body, or
even in a sphere whose radius is supposed infinite. Now each of
these integrals has a definite form which cannot be replaced by
another. It is necessary to make use of it, if we wish to ascertain
the distribution of heat in the body in question. In general, we
could not introduce any change in the form of our solutions, with
out making them lose their essential character, which is the repre
sentation of the phenomena.

The different integrals might be derived from each other,
since they are co-extensive. But these transformations require
long calculations, and almost always suppose that the form of the
result is known in advance. We may consider in the first place,
bodies whose dimensions are finite, and pass from this problem to
that which relates to an unbounded solid. We can then substitute a
definite integral for the sum denoted by the symbol S. Thus it is
that equations (a) and (/8), referred to at the beginning of this
section, depend upon each other. The first becomes the second,
when we suppose the radius R infinite. Reciprocally we may
derive from the second equation (ft) the solutions relating to
bodies of limited dimensions.

In general, we have sought to obtain each result by the shortest
way. The chief elements of the method we have followed are
these :

1st. We consider at the same time the general condition given
by the partial differential equation, and all the special conditions
which determine the problem completely, and we proceed to form
the analytical expression which satisfies all these conditions.

2nd. We first perceive that this expression contains an infinite
number of terms, into which unknown constants enter, or that
it is equal to an integral which includes one or more arbitrary
functions. In the first instance, that is to say, when the general
term is affected by the symbol S, we derive from the special con
ditions a definite transcendental equation, whose roots give the
values of an infinite number of constants.

The second instance obtains when the general term becomes an
infinitely small quantity ; the sum of the series is then changed
into a definite integral.

29—2

452 THEORY OF HEAT. [CHAP. IX.

3rd. We can prove by the fundamental theorems of algebra,
or even by the physical nature of the problem, that the transcen
dental equation has all its roots real, in number infinite.

4th. In elementary problems, the general term takes the form
of a sine or cosine ; the roots of the definite equation are either
whole numbers, or real or irrational quantities, each of them in
cluded between two definite limits.

In more complex problems, the general term takes the form of
a function given implicitly by means of a differential equation
integrable or not. However it may be, the roots of the definite
equation exist, they are real, infinite in number. This distinction
of the parts of which the integral must be composed, is very
important, since it shews clearly the form of the solution, and the
necessary relation between the coefficients.

5th. It remains only to determine the constants which depend
on the initial state; which is done by elimination of the unknowns
from an infinite number of equations of the first degree. We
multiply the equation which relates to the initial state by a
differential factor, and integrate it between defined limits, which
are most commonly those of the solid in which the movement is
effected.

There are problems in which we have determined the co
efficients by successive integrations, as may be seen in the memoir
whose object is the temperature of dwellings. In this case we
consider the exponential integrals, which belong to the initial
state of the infinite solid : it is easy to obtain these integrals1.

It follows from the integrations that all the terms of the second
member disappear, except only that whose coefficient we wish to
determine. In the value of this coefficient, the denominator be
comes nul, and we always obtain a definite integral whose limits
are those of the solid, and one of whose factors is the arbitrary
function which belongs to the initial state. This form of the result
is necessary, since the variable movement, which is the object of
the problem, is compounded of all those which would have existed
separately, if each point of the solid had alone been heated, and
the temperature of every other point had been nothing.

1 See section 11 of the sketch of this memoir, given by the author in the
Bulletin des Sciences par la Societe Pliilomatiqtie, 1818, pp. 1—11. [A. F.]

SECT. IV.] ANALYSIS OF THE PHENOMENON. 453

When \ve examine carefully the process of integration which
serves to determine the coefficients, we see that it contains a
complete proof, and shews distinctly the nature of the results,
so that it is in no way necessary to verify them by other investi
gations.

The most remarkable of the problems which we have hitherto
propounded, and the most suitable for shewing the whole of our
analysis, is that of the movement of heat in a cylindrical body.
In other researches, the determination of the coefficients would
require processes of investigation which we do not yet know. But
it must be remarked, that, without determining the values of the
coefficients, we can always acquire an exact knowledge of the
problem, and of the natural course of the phenomenon which is
its object; the chief consideration is that of simple movements.

6th. When the expression sought contains a definite integral,
the unknown functions arranged under the symbol of integration
are determined, either by the theorems which we have given for
the expression of arbitrary functions in definite integrals, or by
a more complex process, several examples of which will be found
in the Second Part.

These theorems can be extended to any number of variables.
They belong in some respects to an inverse method of definite
integration ; since they serve to determine under the symbols

I and 2 unknown functions which must be such that the result of

j

integration is a given function.

The same principles are applicable to different other problems
of geometry, of general physics, or of analysis, whether the equa
tions contain finite or infinitely small differences, or whether they
contain both.

The solutions which are obtained by this method are complete,
and consist of general integrals. No other integral can be more
extensive. The objections which have been made to this subject
are devoid of all foundation ; it would be superfluous now to discuss
them.

7th. We have said that each of these solutions gives the equa
tion proper to the plisnomenon, since it represents it distinctly

454 THEORY OF HEAT. [CHAP. IX.

throughout the whole extent of its course, and serves to determine
with facility all its results numerically.

The functions which are obtained by these solutions are then
composed of a multitude of terms, either finite or infinitely small :
but the form of these expressions is in no degree arbitrary; it is
determined by the physical character of the phenomenon. For
this reason, when the value of the function is expressed by a series
into which exponentials relative to the time enter, it is of
necessity that this should be so, since the natural effect whose
laws we seek, is really decomposed into distinct parts, corre
sponding to the different terms of the series. The parts express
so many simple movements compatible with the special conditions ;
for each one of these movements, all the temperatures decrease,
preserving their primitive ratios. In this composition we ought
not to see a result of analysis due to the linear form of the
differential equations, but an actual effect which becomes sensible
in experiments. It appears also in dynamical problems in which
we consider the causes which destroy motion ; but it belongs
necessarily to all problems of the theory of heat, and determines
the nature of the method which we have followed for the solution
of them.

8th. The mathematical theory of heat includes : first, the exact
definition of all the elements of the analysis ; next, the differential
equations; lastly, the integrals appropriate to the fundamental
problems. The equations can be arrived at in several ways ; the
same integrals can also be obtained, or other problems solved, by
introducing certain changes in the course of the investigation.
We consider that these researches do not constitute a method
different from our own ; but confirm and multiply its results.

9th. It has been objected, to the subject of our analysis, that
the transcendental equations which determine the exponents having
imaginary roots, it would be necessary to employ the terms which
proceed from them, and which would indicate a periodic character
in part of the phenomenon; but this objection has no foundation,
since the equations in question have in fact all their roots real, and
no part of the phenomenon can be periodic.

10th. It has been alleged that in order to solve with certainty
problems of this kind, it is necessary to resort in all cases to a

SECT. IV.] SEPARATE FUNCTIONS. 455

certain form of the integral which was denoted as general ; and
equation (7) of Art. 398 was propounded under this designa
tion ; but this distinction has no foundation, and the use of a
single integral would only have the effect, in most cases, of com
plicating the investigation unnecessarily. It is moreover evident
that this integral (7) is derivable from that which we gave in 1807
to determine the movement of heat in a ring of definite radius E ;
it is sufficient to give to R an infinite value.

llth. It has been supposed that the method which consists in
expressing the integral by a succession of exponential terms, and
in determining their coefficients by means of the initial state,
does not solve the problem of a prism which loses heat unequally
at its two ends ; or that, at least, it would be very difficult to
verify in this manner the solution derivable from the integral (7)
by long calculations. We shall perceive, by a new examination,
that our method applies directly to this problem, and that a single
integration even is sufficient1.

12th. We have developed in series of sines of multiple arcs
functions which appear to contain only even powers of the variable,
cos a; for example. We have expressed by convergent series or
by definite integrals separate parts of different functions, or func
tions discontinuous between certain limits, for example that which
measures the ordinate of a triangle. Our proofs leave no doubt
of the exact truth of these equations.

13th. We find in the works of many geometers results and pro
cesses of calculation analogous to those which we have employed.
These are particular cases of a general method, which had not yet
been formed, and which it became necessary to establish in order
to ascertain even in the most simple problems the mathematical
laws of the distribution of heat. This theory required an analysis
appropriate to it, one principal element of which is the analytical
expression of separate functions, or of parts of functions.

By a separate function, or part of a function, we understand a
function / (x) which has values existing when the variable x is
included between given limits, and whose value is always nothing,
if the variable is not included between those limits. This func
tion measures the ordinate of a line which includes a finite arc of
1 See the Memoir referred to in note 1, p. 12. [A. F.]

456 THEORY OF HEAT. [cHAP. IX.

arbitrary form, and coincides with the axis of abscissas in all the
rest of its course.

This motion is not opposed to the general principles of analysis;
we might even find the first traces of it in the writings of Daniel
Bernouilli, of Cauchy, of Lagrapge and Euler. It had always been
regarded as manifestly impossible to express in a series of sines
of multiple arcs, or at least in a trigonometric convergent series,
a function which has no existing values unless the values of the
variable are included between certain limits, all the other values
of the function being mil. But this point of analysis is fully
cleared up, and it remains incontestable that separate functions,
or parts of functions, are exactly expressed by trigonometric con
vergent series, or by definite integrals. We have insisted on this
consequence from the origin of our researches up to the present
time, since we are not concerned here with an abstract and isolated
problem, but with a primary consideration intimately connected
with the most useful and extensive considerations. Nothing has
appeared to us more suitable than geometrical constructions to
demonstrate the truth of these new results, and to render intelli
gible the forms which analysis employs for their expression.

14th. The principles which have served to establish for us the
analytical theory of heat, apply directly to the investigation of the
movement of waves in fluids, a part of which has been agitated.
They aid also the investigation of the vibrations of elastic laminae,
of stretched flexible surfaces, of plane elastic surfaces of very great
dimensions, and apply in general to problems which depend upon
the theory of elasticity. The property of the solutions which we
derive from these principles is to render the numerical applications
easy, and to offer distinct and intelligible results, which really
determine the object of the problem, without making that know
ledge depend upon integrations or eliminations which cannot be
effected. We regard as superfluous every transformation of the
results of analysis which does not satisfy this primary condition.

429. 1st. We shall now make some remarks on the differen
tial equations of the movement of heat.

If two molecules of the same body are extremely near, and are
at unequal temperatures, that ivhich is the most heated communicates

SECT. IV.] FORMATION OF EQUATIONS OF MOVEMENT. 457

directly to the other during one instant a certain quantity of heat;
which quantity is proportional to the extremely small difference of
the temperatures: that is to say, if that difference became double,
triple, quadruple, and all other conditions remained the same, the
heat communicated would be double, triple, quadruple.

This proposition expresses a general and constant fact, which
is sufficient to serve as the foundation of the mathematical theory.
The mode of transmission is then known with certainty, inde
pendently of every hypothesis on the nature of the cause, and
cannot be looked at from two different points of view. It is
evident that the direct transfer is effected in all directions, and
that it has no existence in fluids or liquids which are not diather-
manous, except between extremely near molecules.

The general equations of the movement of heat, in the
interior of solids of any dimensions, and at the surface of these
bodies, are necessary consequences of the foregoing proposition.
They are rigorously derived from it, as we have proved in our
first Memoirs in 1807, and we easily obtain these equations by
means of lemmas, whose proof is not less exact than that of the
elementary propositions of mechanics.

These equations are again derived from the same proposition,
by determining by means of integrations the whole quantity of
heat which one molecule receives from those which surround it.
This investigation is subject to no difficulty. The lemmas in
question take the place of the integrations, since they give directly
the expression of the flow, that is to say of the quantity of heat,
which crosses any section. Both calculations ought evidently to
lead to the same result; and since there is no difference in the
principle, there cannot be any difference in the consequences.

2nd. We gave in 1811 the general equation relative to the
surface. It has not been deduced from particular cases, as has
been supposed without any foundation, and it could not be; the
proposition which it expresses is not of a nature to be discovered
by way of induction; we cannot ascertain it for certain bodies and
ignore it for others; it is necessary for all, in order that the state
of the surface may not suffer in a definite time an infinite change.
In our Memoir we have omitted the details of the proof, since

458 THEORY OF HEAT. [CHAP. IX.

they consist solely in the application of known propositions. It
was sufficient in this work to give the principle and the result, as
we have done in Article 15 of the Memoir cited. From the same
condition also the general equation in question is derived by deter
mining the whole quantity of heat which each molecule situated
at the surface receives and communicates. These very complex
calculations make no change in the nature of the proof.

In the investigation of the differential equation of the move
ment of heat, the mass may be supposed to be not homogeneous,
and it is very easy to derive the equation from the analytical
expression of the flow; it is sufficient to leave the coefficient which
measures the conducibility under the sign of differentiation.

3rd. Newton was the first to consider the law of cooling of
bodies in air; that which he has adopted for the case in which the
air is carried away with constant velocity accords more closely
with observation as the difference of temperatures becomes less;
it would exactly hold if that difference were infinitely small.

Amontons has made a remarkable experiment on the establish
ment of heat in a prism whose extremity is submitted to a definite
temperature. The logarithmic law of the decrease of the tempera
tures in the prism was given for the first time by Lambert, of the
Academy of Berlin. Biot and Rumford have confirmed this law
by experiment1.

1 Newton, at the end of his Scala graduum caloris et frigoris, Philosophical
Transactions, April 1701, or Opuscula ed. Castillioneus, Vol. n. implies that when
a plate of iron cools in a current of air flowing uniformly at constant temperature,
equal quantities of air come in contact with the metal in equal times and carry
off quantities of heat proportional to the excess of the temperature of the iron
over that of the air ; whence it may be inferred that the excess temperatures of
the iron form a geometrical progression at times which are in arithmetic progres
sion, as he has stated. By placing various substances on the heated iron, he
obtained their melting points as the metal cooled.

Amontons, Memoires de VAcademie [1703], Paris, 1705, pp. 205 — 6, in his
Remarques sur la Table de degres de Chaleur extraite des Transactions Philosophi-
ques 1701, states that he obtained the melting points of the substances experimented
on by Newton by placing them at appropriate points along an iron bar, heated to
whiteness at one end ; but he has made an erroneous assumption as to the law
of decrease of temperature along the bar.

Lambert, Pyrometrie, Berlin, 1779, pp. 185 — 6, combining Newton's calculated
temperatures with Amontons' measured distances, detected the exponential law

SECT. IV.] LAW OF THE FLOW OF HEAT. 459

To discover the differential equations of the variable movement
of heat, even in the most elementary case, as that of a cylindrical
prism of very small radius, it was necessary to know the mathe
matical expression of the quantity of heat which traverses an
extremely short part of the prism. This quantity is not simply
proportional to the difference of the temperatures of the two
sections which bound the layer. It is proved in the most rigorous
manner that it is also in the inverse ratio of the thickness of the
layer, that is to say, that if tivo layers of the same prism were un
equally thick, and if in the first the difference of the temperatures of
the two bases was the same as in the second, the quantities of heat
traversing the layers during the same instant would be in the inverse
ratio of the thicknesses. The preceding lemma applies not only to
layers whose thickness is infinitely small; it applies to prisms of
any length. This notion of the flow is fundamental ; in so far as
we have not acquired it, we cannot form an exact idea of the
phenomenon and of the equation which expresses it.

It is evident that the instantaneous increase of the tempera-

of temperatures in a long bar heated at one end. Lambert's work contains a
most complete account of the progress of thermal measurement up to that time.

Biot, Journal des Mines, Paris, 1804, xvn. pp. 203 — 224. Eumford, Jlemoires
de VInstitut, Sciences Math, et Phys. Tome vi. Paris, 1805, pp. 106 — 122.

Ericsson, Nature, Vol. vi. pp. 106 — 8, describes some experiments on cooling
in vacuo •which for a limited range of excess temperature, 10° to 100° Fah. shew
a very close approach to Newton's law of cooling in a current of air. These
experiments are insufficient to discredit the law of cooling in vacuo derived by
M. M. Dulong and Petit (Journal Poll/technique, Tome xi. or Ann. de Ch. et
de Ph. 1817, Tome vn.) from their carefully devised and more extensive range
of experiments. But other experiments made by Ericsson with an ingeniously
contrived calorimeter (Nature, Vol. v. pp. 505 — 7) on the emissive power of molten
iron, seem to shew that the law of Dulong and Petit, for cooling in vacuo, is
very far from being applicable to masses at exceedingly high temperatures giving
off heat in free air, though their law for such conditions is reducible to the former
law.

Fourier has published some remarks on Newton's law of cooling in his
Questions sur la theorie physique de la Chaleur rayonnante, Ann. de Chimie et de
Physique, 1817, Tome vi. p. 298. He distinguishes between the surface conduction
and radiation to free air.

Newton's original statement in the Scala graduum is " Calor quern ferrum
calefactum corporibus frigidis sibi contiguis dato tempore communicat, hoc est
Calor, quern ferrum dato tempore amittit, est ut Calor totus fern." This supposes
the iron to be perfectly conducible, and the surrounding masses to be at zero
temperature. It can only be interpreted by his subsequent explanation, as above.

[A. F.]

4GO THEORY OF HEAT. [CHAP. IX.

ture of a point is proportional to the excess of the quantity of heat
which that point receives over the quantity which it has lost, and
that a partial differential equation must express this result : but
the problem does not consist in enunciating this proposition which
is the mere fact; it consists in actually forming the differential
equation, which requires that we should consider the fact in its
elements. If instead of employing the exact expression of the
flow of heat, we omit the denominator of this expression, we
thereby introduce a difficulty which is nowise inherent in the
problem; there is no mathematical theory which would not offer
similar difficulties, if we began by altering the principle of the
proofs. Not only are we thus unable to form a differential equa
tion; but there is nothing more opposite to an equation than a
proposition of this kind, in Avhich we should be expressing the
equality of quantities which could not be compared. To avoid
this error, it is sufficient to give some attention to the demon
stration and the consequences of the foregoing lemma (Art. 65,
66, 67, and Art. 75).

4th. With respect to the ideas from which we have deduced
for the first time the differential equations, they are those which
physicists have always admitted. We do not know that anyone
has been able to imagine the movement of heat as being produced
in the interior of bodies by the simple contact of the surfaces
which separate the different parts. For ourselves such a proposition
would appear to be void of all intelligible meaning. A surface of
contact cannot be the subject of any physical quality; it is neither
heated, nor coloured, nor heavy. It is evident that when one
part of a body gives its heat to another there are an infinity
of material points of the first which act on an infinity of points of
the second. It need only be added that in the interior of opaque
material, points whose distance is not very small cannot commu
nicate their heat directly; that which they send out is intercepted
by the intermediate molecules. The layers in contact are the only
ones which communicate their heat directly, when the thickness
of the layers equals or exceeds the distance which the heat sent
from a point passes over before being entirely absorbed. There is
no direct action except between material points extremely near,
and it is for this reason that the expression for the flow has the
form which we assign to it. The flow then results from an infinite

SECT. IV.] FLOW OUTWARD AND INTERNAL. 461

multitude of actions whose effects are added ; but it is not from
this cause that its value during unit of time is a finite and
measurable magnitude, even although it be determined only by
an extremely small difference between the temperatures.

When a heated body loses its heat in an elastic medium, or in
a space free from air bounded by a solid envelope, the value of the
outward flow is assuredly an integral; it again is due to the action
of an infinity of material points, very near to the surface, and we
have proved formerly that this concourse determines the law of
the external radiation1. But the quantity of heat emitted during
the unit of time would be infinitely small, if the difference of the
temperatures had not a finite value.

In the interior of masses the conductive power is incomparably
greater than that which is exerted at the surface. This property,
whatever be the cause of it, is most distinctly perceived by us,
since, when the prism has arrived at its constant state, the
quantity of heat which crosses a section during the unit of time
exactly balances that which is lost through the whole part of the
heated surface, situated beyond that section, whose temperatures
exceed that of the medium by a finite magnitude. When we take
no account of this primary fact, and omit the divisor in the
expression for the flow, it is quite impossible to form the differen
tial equation, even for the simplest case; a fortiori, we should be
stopped in the investigation of the general equations.

5th. Farther, it is necessary to know what is the influence of
the dimensions of the section of the prism on the values of the
acquired temperatures. Even although the problem is only that
of the linear movement, and all points of a section are regarded
as having the same temperature, it does not follow that we can
disregard the dimensions of the section, and extend to other prisms
the consequences which belong to one prism only. The exact
equation cannot be formed without expressing the relation
between the extent of the section and the effect produced at the
extremity of the prism.

We shall not develope further the examination of the principles
which have led us to the knowledge of the differential equations ;

1 Memoires de VAcadcmie des Sciences, Tome v. pp. 204—8. Communicated
in 1811. [A. F.]

482 THEORY OF HEAT. [CHAP. IX.

we need only add that to obtain a profound conviction of the use
fulness of these principles it is necessary to consider also various
difficult problems; for example, that which we are about to in
dicate, and whose solution is wanting to our theory, as we have
long since remarked. This problem consists in forming the differ
ential equations, which express the distribution of heat in fluids
in motion, when all the molecules are displaced by any forces,
combined with the changes of temperature. The equations which
we gave in the course of the year 1820 belong to general hydro
dynamics; they complete this branch of analytical mechanics1.

430. Different bodies enjoy very unequally the property which
physicists have called conductibility or conducibility , that is to say,
the faculty of admitting heat, or of propagating it in the interior
of their masses. We have not changed these names, though they

1 See Memoires de V Academic des Sciences, Tome xn. Paris, 1833, pp. 515—530.

In addition to the three ordinary equations of motion of an incompressible
fluid, and the equation of continuity referred to rectangular axes in direction of
which the velocities of a molecule passing the point x, y, z at time t are u, v, w,
its temperature being 6, Fourier has obtained the equation

in which K is the conductivity and C the specific heat per unit volume, as
follows.

Into the parallelepiped whose opposite corners are (x, y, z), (x + Ax,y + Ay, z + Az),
the quantity of heat which would flow by conduction across the lower face AxAy,

if the fluid were at rest, would be -K-j- AxAy At in time At, and the gain by
convection + Cw Ax Ay At ; there is a corresponding loss at the upper face Ax Ay ;
hence the whole gain is, negatively, the variation of (-K~,~+ Cwd) Ax Ay At with

respect to z, that is to say, the gain is equal to ( K -^ - C - - (w0) } Ax Ay Az At.
Two similar expressions denote the gains in direction of y and z ; the sum of the
three is equal to (7 — At Ax Ay Az, which is the gain in the volume Ax Ay Az

in time At : whence the above equation.

The coefficients K and C vary with the temperature and pressure but are
usually treated as constant. The density, even for fluids denominated incom
pressible, is subject to a small temperature variation.

It may be noticed that when the velocities u, v, w are nul, the equation
reduces to the equation for flow of heat in a solid.

It may also be remarked that when K is so small as to be negligible, the
equation has the same form as the equation of continuity. [A. F.j

SECT. IV.] PENETRABILITY AND PERMEABILITY. 463

do not appear to us to be exact. Each of them, the first especially,
would rather express, according to all analogy, the faculty of being
conducted than that of conducting.

Heat penetrates the surface of different substances with more
or less facility, whether it be to enter or to escape, and bodies are
unequally permeable to this element, that is to say, it is propagated
in them with more or less facility, in passing from one interior
molecule to another. We think these two distinct properties
might be denoted by the names penetrability and permeability1.

Above all it must not be lost sight of that the penetrability of
a surface depends upon two different qualities : one relative to the
external medium, which expresses the facility of communication by
contact ; the other consists in the property of emitting or admit
ting radiant heat. With regard to the specific permeability, it is
proper to each substance and independent of the state of the
surface. For the rest, precise definitions are the true foundation
of theory, but names have not, in the matter of our subject, the
same degree of importance.

431. The last remark cannot be applied to notations, which
contribute very much to the progress of the science of the Calculus.
These ought only to be proposed with reserve, and not admitted
but after long examination. That which we have employed re
duces itself to indicating the limits of the integral above and below

the sign of integration I ; writing immediately after this sign the
differential of the quantity which varies between these limits.

We have availed ourselves also of the sign S to express the
sum of an indefinite number of terms derived from one general
term in which the index i is made to vary. We attach this index
if necessary to the sign, and write the first value of i below, and
the last above. Habitual use of this notation convinces us of

1 The coefficients of penetrability and permeability, or of exterior and interior
conduction (h, K], \vere determined in the first instance by Fourier, for the case
of cast iron, by experiments on the permanent temperatures of a ring and on the

varying temperatures of a sphere. The value of — by the method of Art. 110,

and the value of h by that of Art. 297. Mem. de I'Acad. d. Se. Tome v. pp.
165, 220, 228. [A. F.]

464 THEORY OF HEAT. [CHAP. IX.

the usefulness of it, especially when the analysis consists of de
finite integrals, and the limits of the integrals are themselves the
object of investigation.

432. The chief results of our theory are the differential equa
tions of the movement of heat in solid or liquid bodies, and the
general equation which relates to the surface. The truth of these
equations is not founded on any physical explanation of the effects
of heat. In whatever manner we please to imagine the nature of
this element, whether we regard it as a distinct material thing
which passes from one part of space to another, or whether we
make heat consist simply in the transfer of motion, we shall always
arrive at the same equations, since the hypothesis which we form
must represent the general and simple facts from which the
mathematical laws are derived.

The quantity of heat transmitted by two molecules whose
temperatures are unequal, depends on the difference of these
temperatures. If the difference is infinitely small it is certain
that the heat communicated is proportional to that difference ; all
experiment concurs in rigorously proving this proposition. Now
in order to establish the differential equations in question, we
consider only the reciprocal action of molecules infinitely near.
There is therefore no uncertainty about the form of the equations
which relate to the interior of the mass.

The equation relative to the surface expresses, as we have said,
that the flow of the heat, in the direction of the normal at the
boundary of the solid, must have the same value, whether we cal
culate the mutual action of the molecules of the solid, or whether
we consider the action which the medium exerts upon the envelope.
The analytical expression of the former value is very simple and
is exactly known ; as to the latter value, it is sensibly proportional
to the temperature of the surface, when the excess of this tempera
ture over that of the medium is a sufficiently small quantity. In
other cases the second value must be regarded as given by a series
of observations; it depends on the surface, on the pressure and
on the nature of the medium ; this observed value ought to form
the second member of the equation relative to the surface.

In several important problems, the equation last named is re-

SECT. IV.] THREE SPECIFIC COEFFICIENTS. 465

placed by a given condition, which expresses the state of the
surface, whether constant, variable or periodic.

433. The differential equations of the movement of heat are
mathematical consequences analogous to the general equations of
equilibrium and of motion, and are derived like them from the
most constant natural facts.

The coefficients c, h, k, which enter into these equations, must
be considered, in general, as variable magnitudes, which depend
on the temperature or on the state of the body. But in the appli
cation to the natural problems which interest us most, we may
assign to these coefficients values sensibly constant.

The first coefficient c varies very slowly, according as the tem
perature rises. These changes are almost insensible in an interval
of about thirty degrees. A series of valuable observations, due to
Professors Dulong and Petit, indicates that the value of the specific
capacity increases very slowly with the temperature.

The coefficient h which measures the penetrability of the sur
face is most variable, and relates to a very composite state. It
expresses the quantity of heat communicated to the medium,
whether by radiation, or by contact. The rigorous calculation of
this quantity would depend therefore on the problem of the move
ment of heat in liquid or aeriform media. But when the excess
of temperature is a sufficiently small quantity, the observations
prove that the value of the coefficient may be regarded as constant.
In other cases, it is easy to derive from known experiments a
correction which makes the result sufficiently exact.

It cannot be doubted that the coefficient k, the measure of the
permeability, is subject to sensible variations; but on this impor
tant subject no series of experiments has yet been made suitable
for informing us how the facility of conduction of heat changes with
the temperature1 and with the pressure. We see, from the obser
vations, that this quality may be regarded as constant throughout
a very great part of the thermometric scale. But the same obser
vations would lead us to believe that the value of the coefficient
in question, is very much more changed by increments of tempera
ture than the value of the specific capacity.

Lastly, the dilatability of solids, or their tendency to increase
1 Reference is given to Forbes' experiments in the note, p. 84. [A. F.j
F. H. 30

466 THEORY OF HEAT. [CHAP. IX.

in volume, is not the same at all temperatures : but in the problems
which we have discussed, these changes cannot sensibly alter the
precision of the results. In general, in the study of the grand
natural phenomena which depend on the distribution of heat, we
rely on regarding the values of the coefficients as constant. It is
necessary, first, to consider the consequences of the theory from
this point of view. Careful comparison of the results with those
of very exact experiments will then shew what corrections must be
employed, and to the theoretical researches will be given a further
extension, according as the observations become more numerous
and more exact. We shall then ascertain what are the causes
which modify the movement of heat in the interior of bodies,
and the theory will acquire a perfection which it would be im
possible to give to it at present.

Luminous heat, or that which accompanies the rays of light
emitted by incandescent bodies, penetrates transparent solids and
liquids, and is gradually absorbed within them after traversing an
interval of sensible magnitude. It could not therefore be supposed
in the examination of these problems, that the direct impressions
of heat are conveyed only to an extremely small distance. When
this distance has a finite value, the differential equations take a
different form ; but this part of the theory would offer no useful
applications unless it were based upon experimental knowledge
which we have not yet acquired.

The experiments indicate that, at moderate temperatures, a
very feeble portion of the obscure heat enjoys the same property as
the luminous heat ; it is very likely that the distance, to which is
conveyed the impression of heat which penetrates solids, is not
wholly insensible, and that it is only very small : but this occasions
no appreciable difference in the results of theory ; or at least the
difference has hitherto escaped all observation.

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