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 . 388-391. Solution of a problem analogous to the foregoing. Different 392-395. The movement of heat in an infinite solid is considered; and the highest temperatures, at parts very distant from the part originally the movement of heat in a ring 397. Second integral (8) of the same equation (a). It expresses the linear movement of heat in an infinite solid 398. Two other forms (y) and (8) of the integral, which are derived, like the 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 401. Notation appropriate to the representation of these developments. The analysis which is derived from it dispenses with effecting the develop- 411. Integral of equation (e) of vibrating elastic surfaces 413. Use of the same theorem to obtain the integrals, by summing the Integral under finite form containing two arbitrary functions of t. integrals 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 the second member of which represents a periodic function 419. The chief character of the theorem expressed by equation (B) consists 420. Use of these theorems in the analysis of imaginary quantities 423. Construction which serves to prove the general equation. Consequences 424-427. The method which consists in determining by definite integrals the unknown coefficients of the development of a function of x under 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 ART. 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 428. General remarks on the method which has served to solve the analytical problems of the theory of heat. 429. General remarks on the principles from which we have derived the dif- 430. Terminology relative to the general properties of heat ERRATA. Page 300, line 3, for A2, A4, ▲ç, read w▲ŋ, w¡, mâg. Page 407, line 12, for do read dp. Page 432, line 13, read (x-a). PRELIMINARY 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, discovered 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 movements 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 complex effects of all the natural forces, and thus has the thought F. H. 1 |