THE

ANALYTICAL THEORY OF HEAT

JOSEPH FOURIER

, TRANSLATED, WITH NOTES,

BY

ALEXANDER FREEMAN, M.A.,

FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE.

EDITED FOR THE SYNDICS OF THE UNIVERSITY PRESS.

CDambntrge : AT THE UNIVERSITY PRESS.

LONDON : CAMBRIDGE WAREHOUSE, 17, PATERNOSTER ROW.

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

1878

[All Rights reserved.]

-•'

k

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

PREFACE.

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

781452

TABLE

OF

CONTENTS OF THE WORK1.

PAGE

PRELIMINARY DISCOURSE 1

CHAPTER I.

Introduction.

SECTION I.

STATEMENT OF THE OBJECT OF THE WORK. ART.

I. Object of the theoretical researches .14

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

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

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

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

experiments 21

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

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

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

VI TABLE OF CONTENTS.

ART. PAGE

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

SECTION II.

GENERAL NOTIONS AND PRELIMINARY DEFINITIONS.

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

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

26. Specific capacity for heat ib.

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

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

30. Notion of external conducibility ib.

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

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

is compound and variable. Luminous heat ib.

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

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

in liquids ^

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

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

heat . ibt

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

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

SECTION III. PRINCIPLE OF THE COMMUNICATION OF HEAT.

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

TABLE OF CONTEXTS. Vll

ART. PAGE

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

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

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

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

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

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

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

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

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

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

SECTION V.

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

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

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

SECTION VI.

THE HEATING OF CLOSED SPACES.

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

m-n^(a-n)

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

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

85, 86. Remarkable consequences of the preceding equation 65

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

Vlll TABLE OF CONTENTS.

ABT. PAGE

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

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

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

v = A + ax + by + cz.

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

same equation 73

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

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

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

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

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

CHAPTER II.

Equation of the Movement of Heat.

SECTION I.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A RING.

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

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

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

TABLE OF CONTENTS. IX

AET. PAGE

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

of the ratio*: We have

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

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

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

dv K d*v 2dv

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

state of the solid 92

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

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

SECTION IV.

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

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

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

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

first section 99

SECTION V.

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

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

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

TABLE OF CONTENTS.

SECTION VI.

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

ART. PAGE

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

v = A - ax - by - cz.

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

were nul 104

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

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

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

has elapsed 109

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

dv K

SECTION VII.

GENERAL EQUATION BELATIVE TO THE SURFACE.

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

dv dv dv h

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

TABLE OF CONTENTS. XI

SECTION VIII.

APPLICATION OF THE GENERAL EQUATIONS.

ART. PAGE

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

SECTION IX.

GENERAL BEMARKS.

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

CHAPTER III.

Propagation of Heat in an infinite rectangular solid.

SECTION I.

STATEMENT OF THE PROBLEM.

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

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

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

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

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

Xll TABLE OF CONTENTS.

SECTION II.

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

ART. PAGE

171 — 178. Investigation of the coefficients in the equation

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

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

£ o O i

SECTION III. REMARKS ON THESE SERIES.

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

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

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

189. Very simple process for forming the series

SECTION IV.

GENERAL SOLUTION.

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

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

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

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

TABLE OF CONTENTS. Xlll

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

ART. PAGE

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

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

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

A =

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

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

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

sin2o; + c

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

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

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

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

7T J 0

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

IT f=3° . r1*

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

2 t=i J o

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

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

*i . A *9« • ' . • ' D.I •• v

xiv TABLE OF CONTENTS.

ART. PAGE

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

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

* i=0 Jo

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

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

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

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

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

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

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

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

and irbt — f dx F(x) sin ix.

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

i=^+co / .,J.jj» /*.X«f X

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

i=— eo \ J —TT •— If J

i=+oo P + ir

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

»=_„•'-«•

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

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

SECTION VII. APPLICATION TO THE ACTUAL PEOBLEM.

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

TABLE OF CONTENTS. XV

CHAPTER IV.

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

SECTION I.

GENERAL SOLUTION OF THE PROBLEM.

ART. PAGE

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

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

of the solution 218

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

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

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

which measures the conducibility 225

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

regarded as arbitrary 228

256, 257. Remarkable consequences of this solution 236

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

the elimination of the coefficients 238

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

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

XVI TABLE OF CONTENTS.

ART. PAGE

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

result 253

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

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

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

dt dx*

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

CHAPTER V.

Of the Propagation of Heat in a solid sphere.

SECTION I. GENEBAL SOLUTION.

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

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

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

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

tan nX

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

values of its roots 268

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

SECTION n.

DlFFEBENT BEMABKS ON THIS SOLUTION.

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

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

TABLE OF CONTENTS. XV11

ART. PAGB

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

time elapsed 286

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

the radius is very small 287

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

of n . ,289

CHAPTER VI.

Of the Movement of Heat in a solid cylinder.

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

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

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

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

i r1* i—

u =— / dr cos (xtjg sin r) ;

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

22 , 2

"f&C>'

the value of the series

c<2 et*

22 22. 42 22. 42. 62

1 t*

is — / dii(f>(tsmu).

irJ Q

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

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

314. Formation of the general solution 301

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

319. General solution 308

320. Consequences of the solution . . 309

XVI 11 TABLE OF CONTENTS.

CHAPTER VII.

Propagation of Heat in a rectangular prism.

ART. PAGE

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

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

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

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

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

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

in the interior of the solid 319

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

CHAPTER VIII.

Of the Movement of Heat in a solid cube.

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

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

335, 336. Formation of the general solution . 324

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

338. Consequence of the solution ib.

339. Expression of the mean temperature 328

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

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

CHAPTER IX.

Of the Diffusion of Heat. SECTION I.

OF THE FHEE MOVEMENT OF HEAT IN AN INFINITE LlNE.

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

fl °° dq cos qx I da F(a) cos ga. 'o

TABLE OF CONTENTS. XIX

ABT. PAGE

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

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

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

I — sin 2 cos qx is i Jo

—

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

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

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

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

34°

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

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

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

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

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

359. The following theorem is proved :

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

The function / (x) satisfies the condition :

348

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

f+» r*

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

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

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

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

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

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

XX TABLE OF CONTENTS.

ART.

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

du d?u

SECTION II.

OF THE FEEE MOVEMENT OF HEAT IN AN INFINITE SOLID.

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

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

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

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

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

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

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

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

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

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

SECTION HI.

THE HIGHEST TEMPERATURES IN AN INFINITE SOLID.

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

TABLE OF CONTENTS.

ART. PAGE

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

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

results of the solution 387

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

SECTION IV. COMPARISON OF THE INTEGRALS.

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

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

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

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

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

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

402. Application to the equations :

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

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

403. Application to the equations :

(/) • • • • 405

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

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

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

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

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

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

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

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

410. Application of the same theorem to the equation

d2v d-v d-v

+ + = ° ...... 41S

xxii TABLE OF CONTENTS.

ART.

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

412. Second form of the integral 421

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

dv dzv

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

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

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

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

418. The same remark applies to the general equation

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

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

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

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

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

dx* dy*

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

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

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

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

TABLE OF CONTENTS. xxiii

ART. PAGB

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

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

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

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

431. Notations proposed 463

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

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

ERRATA.

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

Pages 54, 55, for k read K.

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

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

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

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

r

du 0 (t sin w) =

Page 300, line 3, for Az,