ART.

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

result

274-276. Application and consequences of this solution

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.

.

279. Analytical expression of the two preceding results
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
evidently the most general which can be formed

dv d-v

PAGE

253

255

259

262

= k is

dt dx2

263

CHAPTER V.

Of the Propagation of Heat in a solid sphere.

SECTION I.

GENERAL 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.
e-Kn't, which expresses

sin nx

This remark leads to the equation v=A

the simple movement of heat in the sphere.

infinity of values given by the definite equation

The number n has an

nX

tan nX

=1-hX. The

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.

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

SECTION II.

DIFFERENT REMARKS ON THIS SOLUTION.

294-296. Results relative to spheres of small radius, and to the final tem-

peratures of any sphere.

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 .

ART.

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

time elapsed

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

the radius is very small

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

PAGE

286

287

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

.

294

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

du

dx

and the definite equation is hu+ =0, giving to x its complete value X. 296

311, 312. The development of the function ø(z) being represented by

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

314. Formation of the general solution.

300

301

315-318. Statement of the analysis which determines the values of the co-

efficients

ART.

CHAPTER VII.

Propagation of Heat in a rectangular prism.

321-323. Expression of the simple movement determined by the general
properties of heat, and by the form of the solid. Into this expression
enters an arc e which satisfies a transcendental equation, all of whose
roots are real ..

324. All the unknown coefficients are determined by definite integrals

325. General solution of the problem .

PAGE

311

313

314

326, 327. The problem proposed admits no other solution

328, 329. Temperatures at points on the axis of the prism .

315

317

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

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

in the interior of the solid

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

319

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

335, 336. Formation of the general solution.

340. Comparison of the final movement of heat in the cube, with the

movement which takes place in the sphere

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

329

.

OF THE FREE MOVEMENT OF HEAT IN AN INFINITE LINE.

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:

ART.

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

the variable temperatures.

348. Application to the case in which all the points of the part heated

have received the same initial temperature. The integral

PAGE

333

dq

0 9

sin q cos qx is 1,

338

339

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

The definite integral has a nul value, if x is not included between

1 and -1.

349. Application to the case in which the heating given results from the

final state which the action of a source of heat determines

350. Discontinuous values of the function expressed by the integral

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=-ƒ (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

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

part heated is expressed by an entirely arbitrary function

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

are transformed into definite integrals

359. The following theorem is proved:

360-362. Use of the preceding results. Proof of the theorem expressed

by the general equation:

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

(See Art. 397)

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 same problem may also be solved by means of another form of the

integral. Formation of this integral

365, 366. Application of the solution to an infinite prism, whose initial

temperatures are nul. Remarkable consequences

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

ART.

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

OF THE FREE 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

solves the proposed problem. It cannot have a more extended integral;

it is derived also from the particular value

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

and if at the same time they have the same value for a certain value

of t.

368

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 . . 376

383-385. The same remarks apply to the distribution of heat in an infinite

solid.

SECTION III.

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

382