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Full text of "The analytical theory of heat"

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 WORK 1 . 



PAGE 

PRELIMINARY DISCOURSE 1 

CHAPTER I. 

Introduction. 

SECTION I. 

STATEMENT OF THE OBJECT OF THE WORK. 
ART. 

I. Object of the theoretical researches .14 

210. 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 
1315. 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 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. 

7380. 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 vF (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. 

101105. 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 ; 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 v lt v z , v 3 of three consecutive points gives the measure 

of the ratio*: W e 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 d 2 v 
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. 

126131. 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 

132139. 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. 

157162. 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. 

163166. 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 x z ,y changes according as the value of y 
increases ; x l 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): i V~ (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. 

179181. 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 ...... 

182184. 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 

196199. 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 . . . . 

200204. 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 + 2 5 b + 3 5 c + 5 d + &c. f 
D = a + 2 b + 3 7 c + 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 e x - e~ x . . . ..... . . . 181 

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

^ sin + a 2 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 

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

IT f=3 . r 1 * 

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 r n dacosia\!/(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 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 6 3 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 irb t 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 /*.Xf 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 

238241. 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 

267271. 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 

298300. 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 r 1 * 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 

2 2 , 2 



" f&C> 



the value of the series 

c< 2 et* 



2 2 2 2 . 4 2 2 2 . 4 2 . 6 2 

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 vf(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 d 2 v d 2 v 
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 d s v d z v d s v 
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 

388391. 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 d 2 v . d z v d*v , 

^ = d* + d?-- : "- (e)l nd d? + ^= ...... (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 

d 2 v 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 d z v 

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 = . . . . . .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 (t sin w) = 

Page 300, line 3, for A z , 4 4 , A 6 , read irA^, 7ivt 4 , ^^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 


7380, A- 


66 6 


as 


11 


m n = a 


67 31 


to 


72 4 


j(l + Jf] 


13 


in denominator 


14 


j 


90 16 


V 2 


146 9 


cos (2m - 3x) 


20 


1 




2 2 m 3 


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 6 2 

26 A. 2 

174 30 1 2 2 2 3 2 4 2 5 2 

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 

5 2 

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, 

226 17 a + (a-/3) 



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





14 


within the uracKets 

(7T\ 2 


A 2 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 1317 k 

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

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



_^_ s in 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 dm V 



300 


3 


^9, 


304 


3 


21 





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 


q lt q 2j q^ 










9 


flj, a. 2 , a 3 




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 


Ln T 


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> d P 

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. 206211, 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. 



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 

12 



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 memoirs 1 
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. 
507514, 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 them 1 . 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 known 2 . 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. 350375. 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 

22 



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. 179213. [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 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 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 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 : 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 enclosure 1 . 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.] 

32 



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 w r hich 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 n t by p t 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 1 t 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. 234294, 
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, w r e see that the part of the mass w r hich 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 J5 7 , the coordinates of the point n will be x, y, z + f 
and those of the point n , x, y , z f + % ; 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 B f ; 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 A f 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 x f , 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 

42 



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 + z t 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 ~ b 2 

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 = to x oo , we shall find the quantity 
of heat w r hich escapes from the whole surface of the bar durino- 
unit of time ; and if we take the same integral from x = 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, 4Z 2 A -=- , 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/ 2 X -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 4Z 2 &. ^~ + 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 dx 2 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<l 2 k ^- . 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- 
gesimal 1 ) ; 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 A t 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 v lt v a the tem 
peratures corresponding to the distances x^ x 2 , we have 

v ~{*i-*tk/s -i /2h log v loof v 9 ,, 

~* = 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 l t be the side of a section in the first bar, and 1 2 
iii the second, we shall have, to express the temperatures of these 
two solids, the equations 



Vl = Ae~ 1 and v 9 =Ae~ 

i\, in the first solid, denoting the temperature of a section made 
at distance x lf and v z , in the second solid, the temperature of a 
section made at distance x z . 

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 

x z 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 conducibility 1 , 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 x l and x z 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 

/** /^ -r 2 I* 

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

x * k 2 

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 4K 2 -y- , or kAjkhl*.e * K j 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. 111. [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 

52 



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 a J }, 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, n lt m 2 , n 2 , m 3 , ?? 3 , ??i 4 , w 4 , &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 ( m i~ n i)f an( ^ the quantity which escapes 
c 

from its upper surface is HS(n t m i+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 - n i = ^ ( b ~ 



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



72 THEOKY OF HEAT. [CHAP. I. 

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

rij a = b m 1 . 
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, y f , 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 M f ; further, 
let v be the temperature of m, and v that of m, also let V and 
V be the temperatures of M and M f , 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 (VV) 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 z t 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 z t 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 m t 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 - cK y 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 

rr dv 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> y> z > 

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 w t 
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 dv r , . , , , ,. ,,-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 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 = 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 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 = = ir y 



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 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< i c i en t h measures the external con- 
ducibility, K the internal conducibility, 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 

KSldxdt. 
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 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 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 constants 1 . 

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, VM + 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\ rfjJf 1 



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 v lt V 2 , V 3 , v 4 , &c., the temperatures 
at the points of division whose distances from the origin are 
x v x v x v #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 x 2 x^ of two consecutive points of division, 
we shall have the equations : 



whence we derive the following relation - * = a x + a~ A . 

We should find a similar result for the three points whose 
temperatures are v 2 , v s , v 4 , and in general for any three consecutive 
points. It follows from this that if we observed the temperatures 
v \> v v v s> v v V 5 & 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 a x + 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 

V s $.-; 

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 a x may be derived 

from it by means of the equation a A + 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 a x 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 X 2 . 

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 

^Trx 2 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 

x z --} 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 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 
to temperature 1. Hence it is requisite to divide the quantity 
of heat which accumulates in this shell by 4 r jrCDx 2 dx ) and we 
shall then find the increase of its temperature v during the time 
dt. We thus obtain the equation 

Jr d(x 2 } 
, _ 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>K7rx z -^- 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<7rX 2 . The value of this quantity is 4<h7rX 2 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 w 7 ill 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 dt y 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 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 (^ ld JL\ * T ! 

dt CD \da? x dx) \ 

120. The quantity of heat which, during the instant dt, 
crosses the cylindrical surface whose radius is x t 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 x t 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 x t 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 x t 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/ 



d 2 v 
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 z t 

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 

d 2 v cPv tfv _ 
da?d** dz z ~ 



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 

72 



100 THEORY OF HEAT. [CHAP. II. 

molecules, during unit of time, traverses an infinitely small surface 
G>, situated within the prism, perpendicular to z y 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 CUBE 8 . 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 
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 ~" f1 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 M t 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 u t 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 = lz, and the two planes which bound it are maintained 
by any cause whatever, A at the temperature 1, and B at the 
temperature ; 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, w r hen 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_ /d 2 v dV dV\ 
dt " CD Vdx 2 + 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 ow r n 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 dzdtK -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 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 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) = ; 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(x t 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) = be the known equation to the surface of the solid, 
and v = </> (x, y, z, f) the general equation which ought to give the 

82 



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 ; 

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 Bi a &s 2 =-(m 2 



or a. = ^ &z , denoting by q the quantity (m 2 + n* + p 2 ) " , 

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, d r , 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 : 

aa f 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 y j 



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 

T rdv 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 +p 8 ) , 
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 r 2 = y z + 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 , d 2 v 

~r~ = i r aud ~ra 
dz dr dz dz z 



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* dr 2 \\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 y z + z* = r z , 

dr fdr\* d*r 

yT-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 r 2 , 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*~dr t + r dr 

Hence the equation which expresses the movement of heat 
in the cylinder, is 

dv_J?i(d^) ldv\ 
dt ~~ CD Ur 2 * 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 , d z v d z v 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 d z v_d z v/dr\ 2 dv d~r 



dv _ dv dr , d 2 v __ d*v /dr\ 2 dv d*r 
~ a ~ + 



Making these substitutions in the equation 

dv_Jt_(d*v d*v < 
dt~ CD(dx* + dy z + 

we shall have 

dv K <Pv (dr\* dr\* dz\* dv (d 



The equation x* + y 2 + z 2 = r 2 gives the following results ; 
dr dr z 



dr . fdr\* tfr 

y r ~r~ an d i = I -j- ) + T -;-= 

z 



j 

d-y \dy] 

dr fdr\ z tfr 

z r-^~ and 1 = -^ + r -j-$ . 
dz \dzj dz z 

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 dx z 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 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 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 
K t 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 








>> * 





1 





,, v ... 








1 


The specific conducibility, K ... 


-1 


-1 


-1 


The surface conducibility, h ... 


-2 


-1 


-1 


The capacity for heat, c ... 


-3 





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 
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?~ CDS V 

we find that, with respect to the unit of length, the dimension of 
each of the three terms is ; 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 A t 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 

92 



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 \dx 2 dy 2 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 = 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)_ . 





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 e mx 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 A y 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~" lr cosmy 
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., 
= a sin y + 3b sin 3y + 5c sin 5y + 7d sin 7y + &c., 
= a cos y + 3 2 & cos 3# + 5 2 c cos 5^ + 7 2 c cos 7?/ + &c., 

= a sin y + 3 3 6 sin 3y + 5 3 c 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., 
= a + 3 2 t> + 5 2 c + 7 2 d^ + 9 2 e + H 2 /+ ... &c., 

= a + 3 4 5 + 5 4 c + 7 4 J+9 4 6+ ... &c., 

= a + 3 6 6 + 5 G c + 7 6 ^+ ... &c., 

= a + 3 8 6 + 5 8 c -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., 
= a + 3 2 Z> + 5 2 c +Td +9 2 e +H 2 /+&c., 
= a + 3 4 & + 5 4 c + Td + 9 4 e + ll 4 / + &c., 
= a + 3 6 6 + 5 6 c + Td + 9 6 e + ll 6 / -I- &c., 
= a + 3 8 -f 5 8 c + 7 8 d + 9 8 e + ll 8 / + &c , 
= a + 3 10 6 + 5 10 c + 7 w d + 9 10 e + ll 10 / + &c. 

The five equations which do not contain /are : 
Il 2 =a(ll 2 -l 2 )+ Z > (H 2 -3 2 )+ c(H 2 -5 2 )+ J(ll 2 -7 2 )+ e(H 2 -9 2 ) ; 



0=a(ll 2 -l 2 )+3 6 6(ir-3 2 )+5 6 c(ll 2 -5 2 )+7 6 cZ(ll 2 -7 2 )+9 6 e(ll 2 -9 2 ), 
0=a(ll 2 -r)+3 8 6(ir-3 2 )+5 8 c(ll 2 -5 2 )+7 8 ^(ir-7 2 )+9^(ll 2 -9 2 ). 

Continuing the elimination we shall obtain the final equation 
in a, which is : 

a (ll 2 - 1 2 ) (9 2 - 1 2 ) (7 2 - 1 2 ) (5 2 - 1 2 ) (3 2 - I 2 ) = ll 2 . 9 2 . 7 2 . 5 2 . 3 2 . 1 2 . 

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, 13 2 I 2 , and in the second member 13 2 
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 7 2 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, w r e 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 

n 2 -9 2 n -jT 2 ir- 5 2 ir-3 2 ir-r 
" iv * ~iY~ n 1 ~~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 

II 2 

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 
II 2 13 2 15 2 



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 

5 2 7 2 9 2 

.. &c. 



5 2 -3 2 *7 2 -3 2 9 a -3 2 



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*-5 2> 9 2 -5 2 ir-5 2 " 

We should calculate the value of d for the case of four unknowns 
only, and multiply this value by 

9 2 II 2 13 2 



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 

3 2 5 2 T 9 2 

3* -1 s " 5^T 2 r^V 9^T 2 

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, 






I 2 




I. 






o 


! 2 -3 2 




I 2 


3 2 


c 


r-5 2 


3 2 - 5 2 




i 2 


3 2 5 2 


d 


r-T * 


3 2 - T 5 2 - 7 2 




i 2 


3 2 5 2 7 2 


e 


I 2 - 9 2 


02 Q 2 * (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, 


I 2 


5 2 


7 2 


9 2 


II 2 


u 


r-3 a * 


5 2 -3 2 


72.32 


92 _. 32 


H 2 -3 2 


C 


i 2 


3 2 


7 2 


9 2 


112 &c 




I 2 - 5 2 

I 2 


3 2 -5 2 
3 2 


5 2 


9 2 - 5 2 
9 2 


ir-5 2 " 
n 2 




l - 7 2 


3 2 -7 2 


5 2 -7 2 


9* -7* 


H 2 -7 2 




I 2 


3 2 


5 2 


7 2 


II 2 13 2 




! 2 -9 2 


3 2 - 9 2 


5 2 - 9 2 


7 2 -9 2 


1P-9 2 13 2 -9 2 




I 2 


3 2 


5 2 


7 2 


9 2 13 2 


* 


i 2 -ir 


3 2 -ll 2 


5 2 -ll 2 


* 7*-ll 


* 9 2 -ll a 13 2 -11 2< 



. 



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 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 (2i 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 ; 

102 



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 \x y 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) 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- 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 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 = 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 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 



K 7 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 = 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 ; 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 2ra 8 



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 j c 

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) x t 
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 = 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 principles 1 . 

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 e x ^~ L + arc tan e~ x ^ ~ l j 

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 + -^ = ; 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 ~ = - K 3 * cos 3# ^ =^" 5 * cos 5 y t &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 v l 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 ^irv 2 = 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 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 2lfKj-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 - 7 V = (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 = 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" X 1 

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 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 v t 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 ; 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 + -=-$ = 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 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(x t y) be the equation which represents its 
initial state, and let 1 be the constant temperature of the base 
A y and 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 f v 
transmits, during the instant dt, to the molecule m, a quantity of 
heat expressed by qj.f^ f)dt t the factor q l 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 m z , m 5 , ??? 4 &c. which act on m ; 
that is to say, writing q 2 ,/ 2 or ^ 3 ,/ 3 , or q^ / 4 and so on, instead of 
q v f v In the same manner ^q l (F l 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 q l 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 



112 



, 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 

d 2 v 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 e if ** 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- 

d 2 v d 2 v 
angular solid has led to the equation -y-g + -=- = ; 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 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 + 2 3 6 + 3 3 c + tfd + 5 3 e + &c., 
(7= a + 2 5 ^ + 3 5 c + 4 5 cZ + 5 5 e + &c., 
D = a + 2 7 6 + 3 7 c + 4 7 d + 5 7 e + &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^ a a + 26 2 = A a , a 3 + 2& 3 + 3c 3 = A z , 



3c 4 



3c 5 



&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 &) C 5 , D 5 , E.., &c., we find 



a. (5 2 - I 2 ) + 2\ (5 2 - 2 2 ) + 3\ (5 2 - 3 2 ) 
a 5 (5 2 - I 2 ) + 2 5 5 (5 2 - 2 2 ) + 3 5 c 5 (5 2 - 3 2 ) 
o 5 (5 2 - I 2 ) + 2 7 5 (5 2 - 2 2 ) + 3V 5 (5 2 - 3 2 ) 

We could have deduced these four equations from the four 
which form the preceding system, by substituting in the latter 
instead of 



c 4 , (5 2 -3 2 )c 5 , 

rf 4f (5 2 -4 2 )c/ 5 ; 

and instead of A t , D z A^ B b , 

B t , 5 Jf-C-., 

C 4I 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 



= c s (5 2 - 3 2 ), rf 4 = rf 5 (5 2 -4 2 ), 



&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* - 2 2 ) (4 2 - 2 2 ) (5 2 - 2 2 ) (6 2 - 2 2 ) . . . 



~ (4 a - 3 2 ) (5 2 - 3 2 ) (6 2 - 3 2 ) (T - 3 2 ) . . . 

d = (5* _ 4 ) (G 2 - 4 2 ) (T - 4 2 ) (8 2 - 4 2 ) . . . 

&c. &c (e). 

209. It remains then to determine the values of a lt 6 2 , c 8 , 
d 4 , e e , &c. ; the first is given by one equation, in which A enters; 
the second is given by two equations into which A 2 B Z enter; the 
third is given by three equations, into which A 3 B 3 C 3 enter ; and 
so on. It follows from this that if we knew the values of 

A 19 A 2 B 2 , A 3 B 3 C 3 , Af^CJ),..., &c., 

we could easily find a x by solving one equation, a 2 & 2 by solving 
two equations, a 3 b 3 c 3 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 A 2 in 
terms of A % and 5 2 ; 2nd, by two substitutions we find this value 
of A 1 in terms of A 3 B 3 C 3 ; 3rd, by three substitutions we find the 



SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 173 

same value of A l in terms of J 4 J5 4 (7 4 Z) 4 , and so on. The successive 
values of A are 



A, = A\ 3 2 . 4 2 - B, (2 2 . 3 2 + 2 2 . 4 2 + 3 2 . 4 2 ) + <7 4 (2 2 + 3 2 + 4 2 ) - D 4 , 

^^J^ 2 ^ 2 ^ 2 ^ 2 -^^ 22 - 82 - 4 ^ 22 - 32 - 5 ^ 22 - 42 - 5 ^ 32 - 42 - 52 ) 
+ C 6 (2 2 . 3 2 + 2 2 . 4 2 + 2 2 .5 2 + 3 2 .4 2 + 3 2 .5 2 + 4 2 .5 2 ) 
- D b (2 2 + 3 2 + 4 2 + 5 2 ) + E 6 , &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 

2 2 .3 2 .4 2 .5 2 .6 2 ...&c., 
we have 



" D (.2*. 3". 4" + 2". 3". 5 a + 3". 4". 5" + &C 7 
E .S .^.o 1 + ^~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 

I 2 2" 3" 5 2 6*" 

after having removed the first fraction p. If we represent 

the respective sums of products by P lf Q x , R^ S lt T I} ... &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 



2 2 .3 2 .4 2 .5 2 ... 

CQ l - DR V + ES l - &c., 



174 THEORY OF HEAT. [CHAP. ITT. 

now the quantities P lt Q lf E lt S lt 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 6 2 , c 3 , d 4 , e s , &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 6 2 , the three 
following the value of C 3 , the four following the value of d 4 , and 
so on. 

On completing the calculation, we find by simple inspection 
of the equations the following results for the values of 6 2 , c s , r7 4 , 
&c. 



3c 3 (I 2 - 3 2 ) (2 2 - 3 2 ) = A 3 l 2 . 2 2 - B z (I 2 + 2 2 ) + <7 3 , 
4<Z 4 (l 2 -4 2 )(2 2 -4 2 )(3 2 -4 2 ) 

= .4 4 l 2 . 2 2 . 3 2 -^ 4 (I 2 . 2 2 + I 2 . 3 2 + 2 2 .3 2 ) + C 4 (1 2 + 2 2 -f 3 2 ) -7> 4 , 
&c. 

It is easy to perceive the law which these equations follow ; 
it remains only to determine the quantities A n B n , A 2 B 3 C 3 , 
A$f!v &c. 

Now the quantities A. 2 B 2 can be expressed in terms of A 3 B 3 C 3 , 
the latter in terms of A 4 B 4 C 4 D 4 . 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 I 2 will not enter into the coefficients of the 
value of a t ; that the second 2 2 will not enter into the coefficients 
of the value of b. 2 ; that the third square 3 2 will be omitted only 
from those which serve to form the coefficients of the value of c 3 ; 
and so of the rest to infinity. We have then for the values of 



SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 175 

t> 2 c 3 d 4 e 5 , &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 P 2 , Q,, P z , S 2 , &c., the quantities 

1+1+1+1. 

I 2 3* 4* 5* 



1*. 3 2 I 2 . 4 2 I 2 . 5 2 3 2 . 



&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 b 2 , the equation 

, - &c. 



Representing in general by P n Q n R n S n ... 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 a lt 6 2 , c 
d 4 , e s ..., &c., the following equations: 



A-BP l +CQ l -DB l 



., , 

^- . O . -T . O ... 

A - P 2 + CQ 2 - DR + ES - &c. = 2i ,- " 2 ? 



4 -^= 

l a .2 2 .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 : 

(2 2 - I 2 ) (3 2 - I 2 ) (4 2 - I 2 ) (5 2 - I 2 ) ... 
2 2 .3 2 .4 2 .5 2 ... 

= A-BP 1 + CQi - DE, + ES i - &c., 

(I 2 _ 2 2 ) (3 2 -2 2 ) (4 2 -2 2 ) (5 2 -2 2 )... 
1 2 .3*.4 2 .5 2 ... 

= A-BP,+ CQ. - DR 2 + ES 2 - &c., 

3 2 ) (2 2 - 3 2 ) (4* - 3 2 ) (5* - 3 2 ) . . . 






I 2 . 2 2 .4 2 .5 2 ... 



(1 _ 4) (2 2 - 4 2 ) (3 2 - 4 2 ) (5 2 - 4 2 ) . . . 
I 2 .2 a .3 2 .o 2 ... 

= A - BP, + 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&, P 3 Q 3 ^ 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 
I 2 . 2 2 . 3* I 2 . 2 2 . 4* I 2 . 3 2 . 4 2 2 2 . 3 2 . 4 2 



1 2 .2 2 .3 2 .4 2 2*.3 2 .4 2 .5 2 F.2 2 .3 2 .5 2 > 
by P, Q, 5, 5, &c. 

aj 3 x 5 x 7 
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 P w , Q B , 5 B , /Sf n , &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 P n , Q n , E n , S n , &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 P 2 QA^ &c - 5 those of P &3 R A &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 + 2 3 6 + 3 3 c + 4 3 ^ + 5 s e + &c. = #, 
a + 2 5 6 + 3 5 c + tfd + tfe + &c. = 0, 
a + 2 7 6 + 3 7 c + 4 7 rf + 5 7 e + &c. = D, 
a + 2 9 ^ + 3 9 c + W + 5e + &c. = ^, 
&c., 



SECT. VI.] VALUES OF THE COEFFICIENTS. 

are thus expressed, 



179 



a- A B 

- 



[7 



1T* 1 7T 6 1 7T 4 1 7T 2 

~ + - 



(7 



-lzL 6 ^!^ 4 l 772 ^ x> \ ^ 

[9 2 2 |7 + 2 4 [5""2 6 |3 + 2V~ 



D^-lzL 4 ^!^..!^ 

Vg 3 2 |5 + 3 4 [3 3V 
, F /7r_ 8 _^7r_ 6 ITT* j. 7r 2 . 

1-1 2 * 6 8 " 



-D^.l^ 4 , l^.n 
l7 4 2 5 + 4 4 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 

122 



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? = \7r z - -j= J sin x - (TT* - L= J s i n 2cc -}- ^7r 2 - -^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 

ITT 2 X* 

2 - -T = co 
and integrating we have 



ITT 2 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, 

7T 2 



We could arrive in the same manner at the development in 
series of multiple arcs of the powers x 5 , a?, x 9 , &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 V W + < ^ 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 

7T r 

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 e x 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* n 5 tf ri* + 1 



1 (e* e x ) _ sin x sin 2x sin 3# 

2 71 " 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 x t 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 Cv r /r 

which the foregoing value of 5 must satisfy. 

1 d z s 

Now the integral of the equation s +-5 T~I = </> (#)> m which s 

f ft 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 (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 = 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 = 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 X r -rr AX) sin xir + 2 (5TV r sin 2X r Tr AX) sin 2xir 

r=l r=l 

+ 2 (iT Y r sin 3X r 7r AX) sin 3xir + . . . + 2 (S^Y r sin nX r v AX ) sin nxir 
receives the values F 1} Y^, Y 3 ...Y n corresponding to the values X lt X 2 , X 3 ...X n of 
x, where X r = , 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. 218220 
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 = 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 = 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 = 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 = 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 a lt 2 , a 3 , ... a. y &c., in the 
equation 



< (a?) = a : sin a? + a 2 sin 2# + a 3 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 = 
to X = TT, whence we have 

I <f)(x) sin ix dx = a x 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 a L \ sin ix sin ixdx ; 2nd, that the value of Ismixsmixdx is 
i-TT ; whence we derive the value of a i} 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 = to x TT, in which i and j are integers, is 
jj sin (* - j) x - ^-. sin (i + j) x + C. 

Since the integral must begin when x = 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 

a t \ sin x sin ix da, a 2 1 sin 2x sin ix doc, a 3 (sin 5x sin ixdx t &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 







2 I sin ix sin ix dx == TT ; 



thus we obtain, in a very brief manner, the values of a lt a z , a 3) ... 
4 , &c., namely, 



2 f 2 f 

ttj = - /( (#) sin # dr, a 2 = - 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 (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 and TT ; in this case the integral I sin ixdx is equal to 

9 

?, if the number i is odd, and equal to 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 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 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 x t and which i t \ 
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 



T TT cos x = = s s m 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) a Q cos Ox + a t cos x + a a cos Zx + a a cos 3x + &c. 

+ a i cosix+&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 = 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 = to x IT, supposing j and i to be integers. We 
have 



This integral, taken from x = 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 to TT, we have 

</> (X) COS IX dx = ^TTdi, 

an equation which exhibits the value of the coefficient c^. 

To find the first coefficient , it may be remarked that in 
the integral 

i t 

dn (ji) x, 

if j = and i = 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 = 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 f v [" 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 = a + ttj cos x + a 2 cos Zx -f a 3 cos ox+ ... + a t cos ix + &c., 

and we have, to determine any coefficient whatever a it the equa 
tion a t = 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 a = 7 ?r 2 . We thus form the following series, 

1 A cos x . cos 3# , cos 5% . cos 7x 

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 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 



_ __.._..__..__ 




_&<.. 



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 

5 7r== 2 + rjr375 + o\7- f T9 + 

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^ + ^ ^ "^ + ~ -^ ^ 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 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 = 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 
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 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 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 = to x = IT ; but it is 
sufficient, in the case in question, to take these integrals from 
x = 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. 258262, 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 = 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 term 1 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^ 4 2 ~* ^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 = 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 ) ,, 132 



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. k 2 
2 \ d / 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 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 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) = a x sin x -f <3 2 sin 2# + a 3 sin 3# + . . . -f a t 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 law 1 . 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 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 xdx t &c. 



These integrals, which are taken from x = 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 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 
and 27r r instead of being 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 

Instead of taking the integrals from x = to x TT, we might 
take them from x = 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 to TT is arbitrary; all the other 
parts of the line are determinate. The arc </>(a, which corresponds 
to the interval from 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 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) I n 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 (x) + -^ (a?) instead of <> (a?) : 
for the function cos a? being composed, to right and left of the 

axis of x t 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 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 





In the first equation (P) the integrals might be taken from 
from x = 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 = 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) =A + AI cos 1- ^ 2 cos 2 h^ 3 cos 3 + &c. 

A A A 

. 2-7TX _ STTX ITTX 

+ B^ sin -r + B 2 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 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. 325328. 
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. 404509, 
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. 7577. 
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 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 -- \- a t> e r sin - + ae 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 . . . 

j (x) = a^ sin + & 2 sin -- \- a a sin -- f- &c. 

The coefficients a lt a 2 , 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 r n 

-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 



TTl 1 



- f/W ^ 

- 



We have also 



,-(v+p\/-i) g-to-pV-i) 



J g-(i/+PV-l) 

or 



F(y t p) = *P- 

e v -2cos/?-fe-<" 

cos (# 4- a) - e" 



cos^? 
whence 



2 cos (a? ct) 4- e v e 1 2 cos ^ -L ^ -j- "~ v 
or 



-) + e^] [e v - 2 cos ( 
or, decomposing the coefficient into two fractions, 



TTU = 



fit f -J ^ -i 

J o /() ^ ^_ 2 cos (a? -*) + >-" ~ ^-2cos(^+ a ) + ^j 



142 



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 = 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 to x = TT, and from y 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 d 2 v hi ,7 N 

dt~Cl)dx*~~CDS V 

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~ ht ufx 

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 = : 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 ae mt 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. 

_ k n 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~ kn2t cos nx : here also we have 



2mrr = 2V, 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 ] + a 2 sin (2 *} + &c. 
\ r) \ rj \ rj 

I 

+ & c 



The numbers a , a lt a a ..., 6 , ^, 6 2 ... 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 b 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 , a lt a 2 , &c., 6 , 6 1? & 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 = to 
x = 2-Trr ; we have then 



*) ~ 3 //( 



* 



+ sin 



in (3/ si 



sn 



Knowing in this manner the values of a , 1 , a 2 , &c., 
b , b t , b 2 , &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 = to x = 2?rr. 

The first term ^ \f( x ] d x > 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. 

k 1 

We have seen previously that the permanent temperatures 

of the ring are expressed by the equation v = az x + 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- = must necessarily hold at the point opposite to 

that which is occupied by the source. The condition aoL x boT x = 
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 = ae x + 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 = 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 and TT. It follows from this that we must suppose 
f(x) = 1, and take the integrals only from x = 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 s i n % 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~ ht t-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~ M M. 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 = 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~ M M. 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 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 = a n + (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 x v and # 2 , and their cor 
responding temperatures at the time t by v l and v z ; we shall have 

Vl = Ja + (^ sin^-l-^ cos^-H e ~^^~ ht 

f , t - x * , T, X 2\ - 
v ~ 1 a o + a i 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 a e~ ht , which would remain the same if we chose two 
points situated at the ends of another diameter. The quantity 
a e~ 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 = -\a Q + 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 b lt 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. = a t 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~ ht e~ 



** . , .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> f ITb 

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 

152 



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)> 

J A = -{(s- a )-(a 8 -a 1 )}, 



k 

- 



If we regard a t as a known quantity, we find the expression 
for a 2 in terms of a v and A, then that of a z in a 2 and h ; the same 
is the case with all the other unknowns, a 4 , a 5 , &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 = a x and a n = a^, the value 
of 2 contains the first power of h, the value of a 3 contains the 
second power of h, and so on up to a B+1 , which contains the 
n th power of li. This arranged, a a+l becoming equal to a n , we 
have, to determine h, an equation of the n ih degree, and a t 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* < + a 8 V + &c, 
7 = a/< + ay + a 8 V" -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 n ih degree in h, 
which has, as we shall see further on, all its roots real. 

The coefficients of the first equation a lf 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 a m by the equation 
a m = 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 and then equal to 1, which 
gives a = B sin w, and a l = A sin it, and consequently 

a i / i\ 
a m = , 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 )-<V 2 > 

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 a m being 

-r- 1 - [sin ?m sin (m - 1) w], 
sin u L 

we must have, in order to satisfy the condition a n+l =^ a n9 the 
equation 

sin (n -f 1) u sin u = sin ?m - sin (n 1) u t 

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 a l 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, 

A 7T -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 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. = na l ; 

now the initial values of the variables being a, b, c, &c., we must 
necessarily have 

na l = 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.) + a 1 -- 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 + b t + 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^ b l} q, d lt 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 

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 a iy a z , a Jz ,...a i ...o Jn the initial temperatures 
whose values are arbitrary, and by a v a 2 , a s ...a i ...& 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 a lf a z , a 3 ...a n : 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>) a t + a>a n , (m CD) 2 -f a>z v (in o>) a 3 + coy 2 , . . ., (m a>) a n 
+ 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,, da 2 , (fa 3 ...da.,...cfa H the 



240 THEORY OF HEAT. [CHAP. IV. 

infinitely small increments which the temperatures a 15 2 ,...a 4 ...a n 
receive during the instant dt, we have the following differential 
equations : 



Ja 2 = -dt 

k 
d*i = -dt 

^-i = -^( a - 2 - 2 Vi + <>> 

Illi 

<fe.^ ~ <&(<.._, -2*. 4 a,). 

261. To solve these equations, we suppose in the first place, 
according to the known method, 



The quantities 6 t , 6 2 , & 3 , ... & are undetermined constants, as 
also is the exponent li. It is easy to see that the values of 
cij, ff 2 ,... 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 b 1 ,b z) b 3 ,... 
Z> 4 .,...6 n , 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 b lt b 2> b B ,... b n , 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 b t and h we have 

_2A* . 2JT 

a = sin Oue m " " ^ 



_ verein 

3 = sm zue " " 



a n = sm w 

262. The last equations furnish only a very particular solu 
tion of the problem proposed ; for if we suppose t = we have, as 
the initial values of a 1? 2 , a 3 , ... , the quantities 

sin OM, sin Iw, sin 2u, ... sin (n 1) M, 

which in general differ from the given values a lt a a , a a) ...a n : 
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, a 2 , a 2 , ... a n , were proportional to the sines 

27T -, 2-7T 27T . . - N 2?T 

sm , 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 5 2 , b 3 , ... b n , 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 b t , b z , 6 3 , ... b n , 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 2-7T , . 2-7T 

, 1 , 2 , ..., (w 1) , &c., 

n n 1 n n 

by u lt M S , w s , ..., W B , we can take for the values of b lt 5 g , 6 3 , ... b n 
the w quantities, 

sin Ow 4 , sin lw,., sin 2M 4 , sin 3w 4 , ..., sin (n 1) M, ; 
or else 

cos Qu t) cos lit., cos 2ttj, cos SM,, ..., cos (?i 1) w 4 . 

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 b lf b 2 , b 3 ... b n) 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) w 4 e , a 7i = cos (n 1) M 4 e * 

162 



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 a x , a a , 8 , ... a n 
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 A v B I} A 2 , B 2 , A 3 , B s , ...*A n) B n) any coefficients 
whatever, we may take to express the general value of one of the 
variables, a^j for example, the equation 

/ r> \ ^n vers i n M i 

of wi+l == (A i sin mu l 4 B^ cos muj e 

versin 11% 

+ (A* sin mu>, 4- B cos mu) e " 



-?** versinw,, 

+ (A n sin mu n + B n cos mu n ) e 7< 

The quantities A lt A^A 33 ... A n , J5 X , J5 a , J5 8 , ... B n , which 
enter into this equation, are arbitrary, and the arcs u it u 2 ,u s , ... u n 
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, a a , a 8 , ... a n are then 
expressed by the following equations : 

. _ 

a t = (A l sin Ow t + B l cos OuJ e 5 



sn w + cos 



_ versin 3 

sn w + cos * 



&c.; 



_m versin 



2 = (A^ sin lu^ + B^ cos IttJ e 

- versin w 2 

4 (A 2 sin \u z 4- B 2 cos Iw 2 ) e 1 

~ versin % 

+ (A a sin lu s 4 B 3 cos lnj e 
+ &c.; 



SECT. II.] GENERAL SOLUTION. 245 



a 3 = (A t sin 2t*, 4 B l cos zty 

- versin ?< 2 

4- (^4 2 sin 2w a 4 1? 3 cos 2w 2 ) e 

- ^ versin */ 3 

4 (J. 3 sin 2?/ 3 4 # 3 cos 2w a ) e f 
+ &c. ; 

a n = (^ sin (n 1)^4- ^ cos (n 1) u t ] e m 

versin w 2 

+ [A a sin (n - 1) w a + B 3 cos (?i - 1) u 9 ] e 

- ?** versin 3 

4 {-4. sin (n 1) w a 4 B a cos (*i 1) iij e 
4&c. 

265. If we suppose the time nothing, the values a v a 2 , cr 3 , . . . a n 
must become the same as the initial values a lt a 2 ,a 3 , ... a n . We 
derive from this n equations, which serve to determine the coeffi 
cients A v B V -A 2 , B 2 , A y B 3 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 u l} versin w 2 , 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 , 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 . 2-7T p 

series of quantities versin , 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 u x >, which have the same versed sine, are such that 



246 THEORY OF HEAT. [CHAP. TV. 

the cosine of any multiple whatever of W A is equal to the cosine 
of the same multiple of w A , 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 J5 A + 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 
a i> a 2 > a s " a n> tne fc> re g m g analysis furnishes us with the equa 
tions 

/ . 2-f A 27T\ -* verBinO 2 -? 

a = [A. sin 0.0 H-^ cos 0.0 }e m 
1 \ n n / 

f A 1 27r D i 2lT\ 

+ M 9 sm0.1 +_B 2 cos0.1 }e 

\ n n J 

sin . 2 + B cos . 2 ~ e 

n n 



4 &c., 

,=(A 

+ (* 


sin 1 . 
n 

i i 2?r 
sin 1 . 1 

% 
w 


+ # cos 1.0^ 
?i y 

+ 5,0081.1-1 

71 / 

3 n ) 


. -^versinO 2 ? 

| e w 

-?** versin 1 S JT 

g Hi 

_^ versin 2 ?? 
1 g m 



&c., 



SECT. II.] FORM OF THE GENERAL SOLUTION. 247 



e 



sn 



2.1^ r + 7? 2 cos2.1^)/^ 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 ** 

= j JjSin (n-1) + B i cos ( 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 , 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 A I} B lt A 2 , 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 A V B^A V B V .A^B V &c., 
which enter into the preceding equations, we must consider the 
initial state which is known : suppose t = 0, and instead of 
a lt 2 , 3 , &c., write the given quantities a x , a 2 , a 3 , &c., which are 
the initial values of the temperatures. We have then to determine 
A lf B lt A 9 , B 2 , A a , B 3 , &c., the following equations: 



248 THEORY OF HEAT. [CHAP. IV. 

a x =A 1 sin 0.0^"+ A 9 sin 0.1 + A sin 0.2 + &c. 
n w ?& 

+ B. cos . + jR, cos . 1 + J5_ cos . 2 + &c. 
?i w n 

t . - _ 2?r , - ^ 2?r . ._ 2?r n 

2 = A 1 sin 1 . + 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 = A l sin 2 . 2 - + 4 a sin 2 . 1 + A 8 sin 2 . 2 + &c. 

+ A cos 2 . + 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 


A 3 sin (n 1) 2 h &c. 

5, cos (w- 1)2 + &c. 
w 

,fm\ 



/7T 

w - 1) - 



268. In these equations, whose number is ??, the unknown 
quantities are A lt B lt A 2 , B 2 , A 5 , B s , &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 A i+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)u t 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 B i+l ; we must multiply each 
equation by the multiplier of B i+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 to (n 1), the sum 

2 cos (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 (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 = ; 

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 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 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 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) u t 

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 
A v disappears of itself through having nul coefficients ; to find B^ 
we must multiply the two members of each equation by the co 
efficient of B t in that equation, and on adding all the equations 
thus multiplied, we find 



To determine A 2 we must multiply the two members of each 
equation by the coefficient of A 9 in that equation, and denoting 

the arc - - by q, we have, after adding the equations together, 

W9 

a l sin 0^ 4- a 2 sin Iq + a s sin 2q + . . . -f a n sin (n l)q = 
Similarly to determine B a we have 
rtj cos 0^ 4- a z cos 1 q + a a cos 2</ f . . . + a n 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 : 



-ftf 2 sin I- 77 -fasm2-^" +&c. = 2a i sin(i-l)l 

n n n ^ n 

2?r , 2?r - 2?r 2?r 

-- + GLCOS! +a 3 cos2 + &c.= 2 i cos(z-l)! 

n n n J n 









.2 +a 3 cos2.2 -f &c. = 2a, f cos (i-l)2 
?i ?i } n 

+ 2 sinl.3 + a 3 sin2.3 +&c.=Sosin(*-l)3 

71 71 ?i 

s^^cos 0.3 + 2 cosl.3 +CLCOS2.3 + &c. = 2a i cos(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^=2asin(t-l)(;-l) and ^B =s2aodB(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 A l9 J$ lt A 2 ,B Z , 
A 3 , B 3 , &c.j and the equations (ra), Art. 267, are completely solved. 

272. "We now substitute the known values of the coefficients 
A lt B lt A 2 , B 2 , A 3 ,B S , &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 + JV 2 cos qj 6 <versin ^ + &c. 

= N + (M, sin 2q l + N t cos 2^) 6 * versin * 

+ (M z sin 2g 2 + ^ cos 2g 2 ) 6 < vershl * + &c. 



. = JV + {,, sin ( j - 1) ^ + ^ cos (j - 1) grj e 

+ M sin - 1 + ^" cos - 1 e versin + &c. 



n = i o + sn 71 - q, + , cos n - Sl e 

+ {M z sin ( - 1) q z + JV; cos - 1) 2 } e * versin + &c. 
In these equations 

_ , 27T 27T Q 27T 

e = e , ^y , 2 2 = 2 > ^= 3 &c., 



2 


2^ . 


= ^ a cos (i 1) <7 1} 


M. \ = - 5 a sin 


n 


?^ 


2 V 


2 


= A&t COS (i 1J Q , 


1/ Q = - 2 i sin 


71 


2 ?i 


2 

= S (!< COS (^ 1) Q g, 


2 
1T Q = - 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)~Sasin(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, a v a 2 , a- 3 ... a n , 
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 a v a 2 , a 3 . . . a n 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 

+ - a t cos (j 1) 2 e 

2 . 2?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 ,, a 3 , 4 , ... a n were nul, we should have 

2 + - a 2 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 a 2 , we should find for the value of a j 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 
a jt we suppose the time to be infinite, we find a,- = - 2 a i} 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,- = - 2 t a i 4- - -tain (j -1) 2a f - 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 - 2a f 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 a it 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 a j+l) a j+2) a j+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 . , - - 2a f - = A sin [B 4 Uj 4 1 J at 4- &c. 
n \ n / 

F. H. 17 



258 THEOEY OF HEAT. [CHAP. IV. 



^ +2 _ 2 a . = A sm B + Uj + 2 co* + &c. 
_ _ 2 a . = 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 



172 



260 THEORY OF HEAT. [CHAP. IV. 

As to the initial temperatures a lt a 2 , a 3 ...a n , they depend on 
the value of the arc x, and regarding these temperatures as the 
successive states of the same variable, the general value a t repre 
sents an arbitrary function of x. The index i must then be 

x 
replaced by -y- . With respect to the quantities a lt a g , a 3 , ..., 

these are variable temperatures depending on two quantities 
x and t Denoting the variable by v, we have v = $ (x, t). The 
index j t 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, a it i, a j} /, 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 = 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 - 2a 4 sin (i 1) - 

dx ft n 

is -\ f(x] sin JPC&P, the integral being taken from x = 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 
f n \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# + a z sin 3# + &c. 
^ * ~~ + b t cos x + 5 a cos 2x + b 3 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 and 2?r, without its 
being established so as to apply to values of x which exceed those 
limits. 

279. The function (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?, = 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 = 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 v 1 =f(x) t the fluxion -y 1 
is known, since it is equivalent to k ^ \ . Thus denoting by 

# 2 or v 1 -\-Jc -j- 1 dt, the temperature at the commencement of the 

second instant, we can deduce the value of v 2 from the initial 
state and from the differential equation. We could ascertain in 
the same manner the values v a , v 4 , ... v n 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 v lt v zi v s , &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 v 1 =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 -i z kt 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 

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> = 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. 116131, 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 in t e g r ate 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 e mt u, 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 w 2 . 
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 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 

2 71 " 2 71 " 2 71 " 2 71 " 

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 utsm 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 1TiX 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 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=lhX. 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 n lt n a , n 3 , &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 + a 2 e~ kn & sin w 2 # + a 3 e~ kna2t sin n s x + &c. 

If t be made equal to 0, we have as the expression of the 
initial state of temperatures 

vx = a x sin n t x + a z sin n 2 x + a z sin n 3 x -f &c. 

The problem consists in determining the coefficients a lt a 2 , a 3 
&c., whatever be the initial state. Suppose then that we know 
the values of v from x = to x = X, and represent this system of 
values by F(x) ; we have 

F(x) = - (a x sin n^x + 2 sin njc + a s sin n s x + a 4 sin n^x + &C.) 1 . . . (e). 

2.91. To determine the coefficient a lt multiply both members 
of the equation by x sin nx dx, and integrate from x = 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 w 1 , w 2> w 3 , 

&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. 2527. [A, F.] 

182 



276 , THEORY OF HEAT. [CHAP. V. 

It follows from this that in order to obtain the value of the 
coefficient a lt in equation (e), we must write 



2 \x sin UjX F(x) dx a^\X -^~ sin Zn^X] , 

the integral being taken from x = to so = X. Similarly we have 
2 \x sin n z x 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 and xX determines 
the coefficient a it 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 n z x Ix sin n z x 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 n lt n 2 , 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 = 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 e t , e 2 , e 3 , &c. the roots of the equation 



tan e 

and supposing them arranged in order beginning with the least ; 
replacing n^X, n 2 X, n Q X } &c. by e^ e 2 , 6 3 , &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 e 1 cosec e x 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, 6169. [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 -273 63 

ohX 
or, omitting the higher powers of e, e 2 = ^- . On the other 

hand, the quantity - -- cos e becomes, on the same hypothesis, 

. ex 

27 Y Sm X 

^ 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 n 9 is very much 
greater than ; 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- 

_ * h A. 
duced after the time t to e C DX . 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 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 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 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 v l and v z corresponding to 
the times t t and t z , we have 

hS _ log 0j log v 2 
CDV~ t t -t v " 

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 h r 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 vu 
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 000042 1 ; 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 

ftlog l0 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 Hlog i0 e 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 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 n t 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 ej 2 -fitx* , (sm e, - 6 2 cos e g ) -^P 
.4 e 3 26 - sin 2e * 6 3 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 great 1 . 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 

KTT Z 
being equal to 7 ^y a , 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 determined 1 . 
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 
chapter 8 . 

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 + njl. 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 = 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 = 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 root 1 . 
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 * ^<e t , 

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 d z u 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 

192 



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 9 a 4 2 9 2 4, 2 fi 2 

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- a 2 w 2 e~^ w 4- 3 w 3 e~^ 3< + &c., 

ffi> 9v 9a> & Ct denote all the values of g which satisfy the definite 
equation ; u v u z , u s , &c. denote the values of u which correspond 
to these different roots; a l9 a z , a a , &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 

by / (0) or y, we have 

ffi /9 s 0* 

y =/ (0) = 1 - + * " 



2*. 3 4 a 

the definite equation becomes 

6* O 3 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 ^, ^r 2 , g z , &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(&) 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 = 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 

d l y i,d i+l y ^d i+ *u 

w+v+v &+*%&-* 

d i+l v 
for if we give to 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 = has all its 
roots real and positive. 

309. It follows from this that the equation / (0) = or y = 
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 receives when we give 



to 6 values which continually increase from = to = 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 = lies between two consecutive roots of y = 0, 
and reciprocally. Hence denoting by # t and 3 two consecu 
tive roots of the equation y = 0, and by # 2 that root of the 
equation y = which lies between l and 3 , every value of 6 in 
cluded between l and 2 gives to y a sign different from that 
which the function y would receive if 6 had a value included be 

tween 2 and 3 . Thus the quantity 6 is nothing when 0=0^ it 

y 

is infinite when = 2 , and nothing when 3 . The quantity 

y 

must therefore necessarily take all possible values, from to in 
finity, in the interval from to Z , and must also take all possible 
values of the opposite sign, from infinity to zero, in the interval 

from 2 to # 3 . Hence the equation A = necessarily has one 

i/ 



296 THEORY OF HEAT. [CHAP. VI. 

real root between X and 3 and since the equation y = 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 2 2 .4 2 2 2 .4 2 .6 2 

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 = we 
have the equation 

d^y_ 1 y 
dB i+l i + ldO if 

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 e x ~ 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 , o 3 , a> 5 , &c. are nothing, the same is the case 
with the coefficients of the terms which contain of 1 , o>~ 3 , o>~ 5 , &c. ; 
the coefficient of aT 2 is the same as that of o> 2 ; the coefficient of o> 4 is 



4.6.8 2 2 . 4. 6. 8. 10 ^ 

the coefficient of of 4 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 = 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 
9 U + j + ~ c = 1S -J cos ( ^ sm *) fo* 

the integral being taken from r = to r = TT. Denoting by q this 

[dx 

value of u, and making u = qS, we find S = a + & 2 > an d 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" /9 2 /9* /9 6 

- 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 to u TT, denoting by $ 2 , S# ^ 6 , &c. 
the definite integrals 



we have 



Isirfudu, lsm*udu, I sin 6 u du, &c., 
f fl* fi* f) 6 

(COS (0 Sin tt) <?M = 7T - W $ 2 + rj S 4 - w S t 4- &C., 

j 



it remains to determine $ 2 , ^ 4 , S 6 , &c. The term sin n u, n being 
an even number, may be developed thus 

sin n u A n + B n cos 2u + C n cos ku + &c. 

Multiplying by du and integrating between the limits u = and 
U = TT, we have simply I sin n u du = A n 7r, 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 ~~ 2 2 1 ~~ 2 4 1 2 6 ~ 2 6 l 2 3 * 

Substituting these values of S# S^, S& &c., we find 

1 f 6 Z 6* Q Q 

- J cos (0 sin u) du=I-^ + ^fp - ^ ^ ^ + &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 t 3 

(f> (t sin u) = $ + (/> sin w + - $ sin 2 w + -5 c^" sin 3 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 8 lt $ 3 , $ 5 , &c. are 
nothing. With respect to $ 2 , $ 4 , S R) &c. their values are the 
quantities which we previously denoted by A# A# A R , &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*, x 3 , &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 
T w + -r-a-h- j- = 0. 
K ax x ax 



302 THEORY OF HEAT, [CHAP. VI. 

7)1 X* 

If we make = , and consider u as a function of x, we have 

K u 

du Q d*u 

u + d~e + de^- 

The following value 

_i a 2 J* J* 4 _ & 
u 1 u + a ^2 02 ~r 2 2 3 2 4 2 

satisfies the equation in u and 0, We therefore assume the value 
of u in terms of x to be 

- mo? m* a? m 3 x 3 

~ I 2*" + F 2 2 .1 2 ~ ,77& * :c 

the sum of this series is 



the integral being taken from r = 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 





2 "1-2-3-4- 5- &c. 

i> . Vj m X* 

in which 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 V Z , 3 , &c. ; the 
corresponding values of m are 



2 3 

V2 > Y 2 JT 2 < " * 



thus a particular value of v is expressed by 

_2 2 Atf?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 V one of the roots V 2 , 3 , &c., and 
compose by means of them a more general value expressed by 
the equation 



Z-kf9i r / x \ 

= a l e~ x* I cos f 2 -^ Ju l sin qjdq 



g%# 3 r 
A-a / 



cos f 2^7^ sin c + &c. 



!, a 2 , a 3 , &c. are arbitrary coefficients : the variable q dis 
appears after the integrations, which should be taken from q = 
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 a lf 2 , a z , &c. from the initial state. 
Take the equation 

v = af m ^u^ + a 2 e~ mit u 2 + a/r m ^ u 3 + &c., 

in which w 1? w 2 , w 3 , &c. are the different values assumed by the 
function u, or 

- m x z m* x* 
~ + ~ 



77? 

when, instead of -y-, the values ^, ^ 2 , ^ 3 , &c. are successively sub- 

K 

stituted. Making in it t = 0, w T e have the equation 
V =* a^fj -f a 2 u 2 + 3 w 3 + &c., 

in which F is a given function of x. Let < (x) be this function ; 
if we represent the function u i whose index is i by >/r (xtjff^ we 
have 

^ (x) = a^ (a? V^) + a.^ (x Jg} + a 3 ^ (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 = 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 a l may be .found, all the other integrals 



304 THEORY OF HEAT. [CHAP. VI. 

having nul values. Similarly to determine the second coefficient 
a a , we multiply both terms of the equation 

<f> (x) = a z u^ + 2 w 2 + o 3 u B -f &c. 

by another factor <r 2 dx, and integrate from x = to x - X. The 
factor <r 2 must be such that all the integrals of the second member 
vanish, except one, namely that which is affected by the coefficient 
* a 2 . In general, we employ a series of functions of x denoted by 
"i> "2 s ^ a wn i cn correspond to the functions u iy u# u s , &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, a a , &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 2 w , -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 = 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 = 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 = in the two preceding equations we have 

n n , / a \ in r\ f du da\ 

= C + [ u -} and = D+- r <r w-y-1, 
\ xj a \fc dxj a 

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 = to x = X, we have 

du, 



f d?u 7 fdu da\ fdu da\ f d 2 cr . 

and cr -y-. ax = - 7 a u -j-\ ( 7 a u -y- + lu -=-5 cZa7, 
J ^ \dx dxj a \dx dx] a J dx 2 

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 xj a 



p *(- 

d 2 cr \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^- = in the same manner as the 
K cix dx 

function u satisfies- the equation 

m d 2 u 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~~ = we su PP ose 
cr = xs ; as the result of this substitution we have the equation 

n d*s Ids _ 
k S *~d a ? + 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 m v m^ m 3 , &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 + a a w- a + o,w 8 + &c., 

and integrated each side from a? = 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+.-+-P l or ^4-- 

T 



202 



308 THEORY OF HEAT. [CHAP. VI. 

and also lix ^ + ^x^ f = 0, 

or, hty + pfy = ; 
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 U t 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 



m t or 



\319. It follows from the foregoing analysis that we have the 
, two equations 

! x f-, fhX\*}X*U* 

b = 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) =a 1 u l + a 2 ii 2 + a 8 u a + &c., in 
which the coefficients a v a 2 , a 3 , &c. are to be determined, we shall 
find the coefficient denoted by a. by multiplying the two members 
of the equation by xu t dx, and integrating from x = 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 a t . The coefficients a l9 a 2 , a 3 , . . . a p being 
thus determined, the condition relative to the initial state expressed 
by the equation <f> (x) = a^ + a 2 u 2 + a 3 u s + &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? = to 
x X, and to. find the function u wer must integrate from q = to 
<2 = 7r; (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 6 V 6 Z , 6 y &c. are the real and positive 
roots of the equation 

J L X -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 lt 2 , V &c.; for this reason, supposing the roots to be 
arranged according to their magnitude, and 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 small 1 , we find that 

2ft 

the temperatures decrease in proportion to the fraction e" CDX. 

If on the contrary the radius X is very large 2 , 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 temperature 3 of the cylinder whose radius is X y 
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 T f 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 

_ &kt e 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= _ w ill 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) = is 1-4467, neglecting terms after 4 . 

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 
m z n 3 p z 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 I v + 2~ = ^ when y = l or ~Z, and the equation 

k V + ~dz = Wll6n Z = l r ~ l ( Cna pter 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 e lt e 2 , e a , &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 w 2 = n 2 + p 2 . 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 a lf a 2 , a 3 , &c., 7> 1? 6 2 , 6 8 , &c., con 

stant coefficients, the value of v may be expressed by the following 
equation : 



v = (a l e~ x % 2 +% 2 cos njj -f a a e" a ?+^ cos njj + &c.) \ cos n^z 
4- (a^e~ x ^ + n ** cos n$ -f a -* ****+"* cos njj + &c.) 5 2 cos n^z 
+ (a^-* V ^ 2+W 3 2 cos n 4- af-****+* cos n z y -f &c.) b a cos n 3 z 
+ &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 and I. Now, on making 
x 0, we find 

v = (a t cos n^y + a 2 cos n z y + a 3 cos n 3 y + &c.) 
x (^ cos n^z 4- > 2 cos n z y -f & 3 cos n z y + &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 n z y + a 3 cos ?? z y + &c , 
1 = \ cos n$ + b 2 cos v 2 y + & 3 cos njj + &c. 

It is sufficient then to determine the coefficients a lf a a , 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 n lt n 3 , n s , &c. form the 
series of odd numbers (Chap. III., Sec. IL, Art. 177). Here 
?ij, n 2> n 3 j &c. are incommensurable quantities given by an equa 
tion of infinitely high degree. 

324. Writing down the equation 

1 = dj cos n^y + a a cos n$ + a 3 cos n. A y + &c., 

multiply the "two members of the equation by cos n^y dy, and take 
the integral from y = to y l. We thus determine the first 
coefficient a r 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 ny t the integral 



a Icos ny cos vy dij or ^al cos (n v) y dy + -^ a /cos (n + v) ydy, 



sin (n " ")* + ^T V sin (n +v] 

and making y=-l t 

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 n l cos vlv 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 = a i cos 71$ + 2 cos n 2 y + a s cos n z y + &c. 

we wish to determine the coefficient of a term of the second 
member denoted by a cos ny y we must multiply the two members 
by cos ny dy, and integrate from y = to y L We have the 
resulting equation 

f l * * 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^ a 2 , a 3 , &c. may be determined ; the same is the case 
with b lt 6 2 , 6 3 , &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 d zv 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 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 n z y sin n s l cos n z y 
in 2 2n7 + sin 2 2w^ + sin2w C J 



1 _ sin n cos n 
~ 



CHAP. VII.] THE SOLUTION. 315 

or denoting by 6 1} e 2 , e 3 , &c. the arcs nj., n t l, n 3 l, &c. 

e.y . e 9 y . ey 

sin e, cos -~ sin e 2 cos -~- sin e 3 cos -y- 

1 



+ _ + & c . 



4 2e x + sin e l 2e a + sin e 2 2e 3 + sin e 3 

an equation which holds for all values of y included between 
and I, and consequently for all those which are included between 
and I, when x = 0. 

Substituting the known values of a l9 b lt a a , & 2 , a a , b 3 , &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?i 2 Z V 2^? + sin 2n^ 



sin w ? cos n.z f sin w.Z cos n. y 

j __ s _ _ _ I _ i ~ g a 

2/i 3 ^ + sin 2n 2 l \ZriJ + sin 2/i^ 
+ &c .................................................... (E). 

The quantities denoted by n lt n^ n B , &c. are infinite in 
number, and respectively equal to the quantities j , j , , 3 , &c. ; 

the arcs, e 1 , e 2 , e g , &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~> 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, y t 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 ; 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 

o 71 " o 77 " 9 71 " &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, n l9 n z , w 3 , 
&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 n z , 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 e 2 = -r ; the first value of 



CHAP. VII.] CASE OF A THIN BAR. 319 

e, or e lf is \J j- ; by inspection of the figure we know the values of 
the other roots, so that the quantities e lt e 2 , e 8 , e 4 , e 6 , &c. are the 
following A/ -j- , TT, 27r, STT, 4-Tr, &c. The values of n v n v n 3 , n^ n y &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 = to z I, the half thickness of the bar, and then from 
y = 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 y t 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 = to z = l, and a 
second time from x = 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 m 2 + 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 (m z + ?i 2 ) 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;?i 2 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, n a l, 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~ mt cosnx 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(n z + p* + q 2 ), the preceding value of v will always satisfy the 
partial differential equation. We have therefore the equation 
v = e - k (n*+ P * + q 2 )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). 



212 



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 n t 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 n v n z , n 3 , &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 = (a t cos n^x e~ kn & + a 2 cos n z x e~ kn ^ + a 3 cos n 3 x e~ *" & + &c.), 
(b l cos n^y Q-IM + ^ cos n ^ e -kn?t _j_ 3 CO s n$ e~ kn * H + &c.), 
(Cj cos n^z e~ kn ^ + c 2 cos n 2 z er*"** + c 8 cos n s y 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 
a x , a 2 , 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 a 1} a 2 , a 3 , &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 + a 2 cos n z x + a a cos n s x + &c., 
1 = & x cos n t y + 6 a cos n 2 y + b 3 cos n^y + &c., 
1 = c l cos n^z + c a cos n z z + c a cos n B z + &c., 

in which it is required to determine a lt a t , a s , &c. After multi 
plying each member of the first equation by cosnx, integrate 
from # = 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 n z a 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 n s x -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 n n a 

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 : 

, ; 



cos 



CHAP. VIII.] ONE SOLUTION ONLY. 327 

n l} w 2 , ?i g , &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> (y y 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 n v n^ n 3 , &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 > s i nce the three complete 
integrals have a common value, hence 

e-^+ Ac. 



V nfl J PI \ n t a 

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 6 1} e a , e 8 , &c., 



CHAP. VIII.] CUBE AND SPHERE COMPARED. 329 




6, is between and - TT, e 2 is between TT and , e 3 between 2?r and 

- TT, the roots e 2 , 6 g , e 4 , &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 2e l5 2e 2 , 2e 3 , &c., 
are included between 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 

16 1 ^ 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 -^ = e 2 , 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 a 2 
and a z 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 t t 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 
n 3 : n 3 , if the corresponding changes of temperature are always equal we must have 

n?n dt _ n 2 ndt 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 ; 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 fd z v d*v d z v\ , . 

dt~ C7)(dx 2+ d^ + dz 2 ) 

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 ; 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~CDdx z CDS V 

(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~ ht u, and we have 



du -, d*u 

dt dz* Jc v 

\ 

Assume a particular value of u, namely, a cos qx e"^ 1 ; a and q 
being arbitrary constants. Let q v q 2 , q 3 , &c. be a series of any 
values whatever, and a l9 a 2 , a 3 , &c. a series of corresponding 
values of the coefficient Q, we have 

u = a l cos fax) e~*<zi 2< + a 2 cos faai) e~ kq ^ + a a cos fax) e-^* + &c. 
Suppose first that the values q lt q^, q s , &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^ a 2 , a 3 > &c. are pro 
portional to the ordinates Q of the same curve, and that they 
become equal to Q^dq, Q^dq, Q 3 dq, &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 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 = 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 Q t cos qjc + dqQ z cos q z x + dqQ 3 cos q z x + &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 = to x mr, where n is an 
infinite number, and r represents a magnitude equal to any one 
of q lf q z , q 3 , &c., or which is the same thing dq, 2dq, 3dq, &c. Let 
q i be any value whatever of the variable q f 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 = 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 = to a = oo . We have then 

!L_ _. e -u I fa cos g X 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 =k d J ?~ 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 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 , 

= ---- * 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 



222 



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 e x when x is negative 1 . 

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~ hf u we have 



_ , 
dt ~ dx*> 

the general value of u may be expressed as follows : 

u = a i e~ k9iH sin gjc + agr*^ sin gjc + a & e ~ *0& sin g a x -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 -^ + a 2 e sin =- + &c. 

It remains only to find the series of constants a lt a a , a 3 , &c. 
Making t = we have 



. . . 

sin -.+ a sin -- + a 3 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 + a a sin 3r -f &c. 

2 r 

Now, we have previously found a = - \drf(r) sinir, the inte 
gral being taken from r = to r = TT. Hence 



The integral with respect to x must be taken from x = 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 -- , 



jpi 2 * . 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 = 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 = to x = oo. 
We may also write 



TTl) f 30 f 

_ Q-U \ dqe-Wt sm q x I 
* Jo Jo 

7TV f 30 f 30 

~^ 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 parts 1 . 

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 f 30 f 00 

-_ = 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~ kqH s 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^ CO s 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 
= 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- M l dqe~ k< * H dx (a) cos ^ ( a), 

JO J -oo 

r + oo Too 

or, 7rv=e~ ht \ dz<l>(oL) I dqe- k * 2t cosq (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 = 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 = 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 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 > 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 integral 1 - 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 expressions 2 

2 r dq cos qx , 2 f qdq sin qx t 
vJt 1 + <f FC W 1 + 2 2 

the first (Art. 350) is equivalent to e~ x when x is positive, and to 
e x when x is negative. The second is equivalent to e~ x if x is positive, 
and to e x 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 and TT, and its value is 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 and TT, the original equation being true for values of x between 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 = to u = TT, hence the integration with respect to 
x must be taken from x = 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 djnnpj 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)=x r , 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 -^ldusmuu r , the integral being taken from u 
nothing to u infinite. 
Let fjL be the integral 

00 

du sin u u r ; 
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 
x r fjiv. We must then have, according to the condition expressed 
by the equation (e), 

| 7T Of = fJLV X f Or JJLV 7T J 

thus the product of the two transcendants 

/*, r . , [ x du ._ . . . 

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 following 1 , 

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 infinite 1 ? 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 = 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^ g i n %% 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 lastly 1 , 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 
after the integrations. 

Boole, On the Analysis of Discontinuous Functions, in the Transactions of the 
fioyal Irish Academy, Vol. xxi., Dublin, 1848, pp. 126130, 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. 232244, 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 quantity 1 . 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) (E f ) 

TfJ JO 

which differs from equation (E) in that the limits taken with 
respect to a are 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 (E 1 ) 
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 nothing 2 . 

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 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~ x j 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~ ht u we have -,- Jc j- a : the value of u or 

dt dx a 

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>J T ^ 

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 e kt . 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 6 2 = kt 



hence the preceding value of u or e~* e kt 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 a lt a 2 , a 3 , &c., and n v n z , n s> &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 2 r = 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 

232 



356 THEORY OF HEAT. [CHAP. IX. 

given by M. Laplace 1 , 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 = 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 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 

= to 



1 Journal de TEcole Polytechnique, Tome vm. pp. 235244, 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~ Xl v^s ffu r 

7= ecus \dre- r \ 

VTT J 

this integral idre-^ must be taken by hypothesis from 



* + 2 2y "^ ==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 

r Kt 
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 e CDS dre" e~ w ^ e ona 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 difficulty 1 . 

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 (7?) the integral ^ \dre~ r * taken from r = 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 (R) we have 




Paris, 1826. 4to. pp. 5201. 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~ tz dt 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 = to r = oo , 

- 

and consequently is equal to \J-jr; therefore v is nothing; 3rd, 

different points of the solid situated at different depths cc lt x v # 3 , 
&c. arrive at the same temperature after different times t lt t it t & , 
&c. which are proportional to the squares of the lengths x lt a? 2 , x z , 
&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~ ht z, 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 

_ X J* 
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 = 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^e x \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 = ; 



denoting again by ^{R} the integral ldre~ rZ taken from r = 
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~ w2 w 2m 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 /, u 2 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 e kt , or in general u e~ n ? e n 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 * q n . 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 + a a e~ n ** kt cos n B x + &c. ; 
or this, 

u a^e" n ^ kt sin n^x -h a 2 e~ n ** kt sin n z x + a & e~ n * lkt sin n a x + &c. 

a,, a 2 , a s) &c,, and Wj, w a , n B , &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 -*0 2 + 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 er n2 *< sin w^ = J ^ e~^ sin n(x + 2q V^) , 
we could determine in the same manner the integral 

,+ 

I c?2 e-3 3 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 w 8 a; + 6 2 cos w^) 

+ e-"" 2 ^ (a a sin w 3 ic -f 6 3 cos up) -f &c. 
is equivalent to 

-i Cdq 9 ~* I* 1 Sin Wl (iC + 2 2 V ^) + a a sin w, ( 4- 2^ VS) + &c. | 

v/7rj. (^ cos Wl (a: + 2j V^) + 6 8 cos 7i a (x 4- 2 2 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 forms 1 . 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. 170174. [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 ,-* *3L f - 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 = TT 3 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(x y 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(x t 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 



IT 9 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~ nH cosnx, 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~ nZt cosnx 



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 

242 



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 

_1 J 

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 
^ = j 2 , 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 integral 1 



/ 

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~^ 2 cos 2pu, by assuming r?t =p 2 and 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--p a - 

~ 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~ nH cosnx. 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^ + dy z + 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 = 2 3 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 = in order to ascertain the initial state, we find 

3 

v = 2 3 ^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 X y 
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 

X 2 

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 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 = 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 w hi cn f rms the exponent reduces in fact to jy- ; that 

( a _ xf x z 
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 e y 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 _^_ r g 

&/(*)* ** 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 and x = X, may be represented very 
nearly by the reduced equation 




380 THEORY OF HEAT. [CHAP. IX. 

and that 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 



a 2 \ 
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 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 _ ft 2 

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, + Y f +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 _|_ 2 2 \ 

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) i ft) 2 Ct) 3/ i g 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 e H+ * becomes equal to e M , 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* - ft 2 - 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 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 

Tpi 

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 



__ ff CD 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 ; w T e 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 = to x = b. 

We now suppose that the same quantity of heat If is contained 
in an infinitely small element, from x = to x = a). The tempera- 

252 



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 x z x 2 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^ an d ^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 



w e - 



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?; 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~* K 9 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 have 1 

K 9 




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 H t 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 x z 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 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 =fdajdbfdo e - ^ -/(o,M ......... (E). 

The integrals must be taken between the limits 



The limits a lt + a 2 , b lt + b 2 , c 1 , + c 2 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 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+j da 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 r 2 , 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 ^i ca 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 

-a 2 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 r 2 , 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 7rr s , 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 = 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, - 3 5 &c., 

and instead of cos 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+2pJt) 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" + ldtv iv , and continuing always similar substitutions, we 
find 

v = c+ jdt v" 



\c" +jdt (c iv +jdt vJ] , 



or v = c + tc"+~d v + ^G + ^c + &c ............. (I 7 ). 

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 v t ; 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 v ti its value a tl + b^x + Idx Idx v 4llt and so on. By continued 
substitutions 

v= a + bx + \dx Idx v t 
= a + lx+ \dx\dx \a t + Ix 4- Idx Idx v J 
= a + bx+ldx \dx a t + bx + Idx Idx (a u + b t x -f \dx \dx v\ | 



SECT, IV.] NUMBER OF ARBITRARY FUNCTIONS. 401 



or t; = a + -^ a t + 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 = e tD<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 = e tl} * <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). 

262 



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(x y y> f) of three variables. If we make t = 0, we 
have/= (a?, y, 0) = < (a?, y) ; and denoting ^/fe y, 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 (ZD 2 ) (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 -^ + D z v = 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* dx 6 



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 t t would be determined. To express v, we 
should have therefore the equation v e tj) ^ (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.* + ca 6 + 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 de tD , N _. d*v , d*v d*v p 
-T: = ^r 9 () = -Dv = a -j t + b -, 4 + c -j 6 + &c. 
c?^ ai fic 2 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 d 4 v d e v 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* + rp 6 + &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 
plate 1 , 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 d 4 v ,, . , 

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. 169177. [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(x t 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 q 1 *), 



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= 27rj d *^ W fa - 2 sin & cos (<l x - 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 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 = 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 AiT 1 / \ / 

(k/ \ **/ 



From this we conclude 



Denoting j- by another unknown p, we have 



a = x + 2/,6 Jt, da. = 
Putting in place of sin (^ + A fc2 J i ts 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 ce 2 , an d I dx sin x* ; 

f-f-oo r-f-oo 

g = I dx cos cc 2 , 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 + b 2 ), h = ( ^ dy sin (y 2 + 2by + 6 2 ), 

J 00 J 00 



= f di I cos ^ cos 2 ^ cos ^ ~~ cos ^ 2 s * 

J I sin y 2 sin 2by cos 6 8 - sin y 2 cos 2by sin b 2 ) 

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 6 a I dy cos y z 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 y 2 sin 2by cos b 2 sin y 2 sin 26y sin 
and, omitting also the terms which contain sin 2by, we have 

h - cos & 2 J dy sin y 2 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 y z cos 2&# and \dy cos ?/ 2 cos 2% 

which we shall denote respectively by A and B. We may now 
make 



sn cos >2 = 



y z = p z t, and Zby = pz ; or i 
we have therefore 

fj"t\dp cosp*t cos)2 = A, *Jt\dp si] 

The values 1 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 



= r 1 \ dy e "^ = f 1 \ 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?2sins 2 =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 = \dy cos y* \dy siny 2 , 



or 



It remains only to determine, by means of the equations (a) 
and (6), the values of the two integrals 

I dy cos y z cos 2by and | dy sin y* sin 2by. 
They can be expressed thus : 

A = I dy cos y* cos 2by = h sin 6 2 + g cos 5 2 , 

B = I dy sin i/ 2 cos 26^ = h cos 6 2 - ^ sin b 2 ; 
whence we conclude 



writing sin ^ , or cos ^ instead of i/ - , we have 

= ^ s ing4-|) .................. (1) 

and I dpsmtft cospz= -ILsmt 7 ^^-}.., ,..(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- / d P\ 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 = in the value 

S/ IJ 

of -- y or f (x,y y 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 

( 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 - 








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 = 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 e mz , 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 

( 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 (P x -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*~ 

272 



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 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 = tf v = 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 (p z t + 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 = in u and in IF, the first function becomes 
equal to $(&,y), and the second nothing; and if we also make 

= 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 + 2vji t 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 p 2 , +p*, p 6 , &c. 

We have then, on writing once the common factor cos (ptpz), 



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 , p 4 ^ p 6 ^ 8 , i <2*y 5 

^=-/+- or s? = ~ 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> dx 2 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~v x ) 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 = 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 = 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 f t 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 heat 1 ; 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. 2068. [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 2 2 , 4 2 , 6 2 , &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 w r e 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 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 = 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 w r hen 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 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 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 and X. The sign 2 affects the 
integer number i t 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 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 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 w r as 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+Jlty. 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~ p a /( a ) I 



cos ~* e pv + 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. 

282 



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 = in it, it gives v =f(x) ; 3rd, that on making y 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 

d l [* 

i th order, -T-j/OOi o r of the integral I dx l f(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 Sciences 1 , 
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 7 1 ) 

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 



J 



r sin J r > 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 fi lt //, 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 \> a v a s - - - a i I they nav e 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 a l} a a , 
r/- s ... a i? in the equation 

ocF(x) = a t sin (^x) + a 2 sin (JLL^X) -4- a 3 sin (p 3 x) + &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 real 1 . 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 a lt # 2 , a g ,...a f , &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 and the 
whole length X. 

To find the coefficient ., we have multiplied equation (e) by 
dx sin fi t a;, 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 p i 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 a it 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 a t 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 = 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 # = 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), f r (x], f r -i(x), are three successive differential coefficients of f(x), the values 
of x which make f r ()=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 = 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 = to x = X, the same value of 
a n 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 
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 and X. In general the resulting equation 
(e) ought to be applied to values of x, included between and ^Y. 
Now the process which determines the coefficient a t does not 
explain why all the roots ^ must enter into equation (e), nor 
why this equation refers solely to values of a:, included between 
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, a v a 2 , 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, f n in the n ih . To determine the first coefficient a lt we 
multiply the first equation by a- lt the second by cr 2 , the third by 
<7 3 , and so on, and add together the equations thus multiplied. 
The factors <7 1} cr 2 , o- g , ...o- tt must be determined by the condition, 
that the sum of all the terms of the second members which contain 
a a must be nothing, and that the same shall be the case with the 
following coefficients a a , c& 4 , ...a n . 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 p l , p 2 , p 3 ,...p n respectively, 
and determine these factors so that on adding the n equations, all 
the coefficients may be eliminated, except a 2 . We have then an 
equation to determine a 2 . 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 and X. The series <r l , cr 2 , <r 3 ,...<r n 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 a it 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 a lf 2 , a a ...a n , 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 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 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 p lf /z 2 , ^ 3 ...^, &c. will be integers, and the con 
dition 

I ec 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 a t , b iy 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 /, f 2 ,f s ---f n 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 
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 fi f x 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 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 f lt f 2 ,f s .../ 
was expressed by a function subject to a continuous law. such as 
x or x 3 , 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 = to x = X, 
coincides then in this interval with the curve whose ordinate is 
</> (x), and the coefficients a lt a 8 , a 3 ... a n of equation (e) determined 
by the preceding rule always satisfy the condition, that any value 
of x included between 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 bx 2 ; and 
applying to this function the rule which determines the coeffi 
cients, bx* would be developed in a series of terms, such as 

a l sin fax) + a 2 sin fax) + a z sin fax) + ... + a n 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 
bx z , denoting a series of values given for the interval from 
to X, can be developed in a series of terms, such as a t 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 (/i 4 0) 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 
and X. 

If we supposed the primitive heat to be contained in a part 
only of the solid sphere, for example, from x = 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 = 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 a t) and that we have 
formed the equation 

x sin x = a t sin JJL^X + 2 sin JJL Z % + a 3 sin JJL^X -f &c. 

It is certain that on giving to x any value whatever included 
between 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 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 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 and A 7 ". With respect to the 
existence of functions, w T hich 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 (/i 4 ot) aF(a) between the limits a r and 

a = r + 2w. More generally, equation (E) being capable of being 
put under the form 



f x j -vi \ 
v = I ay. . ctr (a) sin /^a 





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. 

292 



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 integrals 1 . 

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. 111. [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 sufficient 1 . 

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 experiment 1 . 

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 radiation 1 . 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. 2048. 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 mechanics 1 . 

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. 515530. 

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 permeability 1 . 

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 temperature 1 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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