which I have since made a great number of applications, and which I shall also constantly follow in this work. The Sixth Chapter contains already the application to the general equations of the motion of heat in the interior and on the surface of a body of any form either homogeneous or heterogeneous. It leads in every case to two remarkable equations, one of which serves to determine, independently of one another, the coefficients of the terms of each series, and the other to demonstrate the reality of the constant quantities by which the time is multiplied in all these terms. These constants are roots of transcendental equations, the nature of which it will be very difficult to discover, by reason of the very complicated form of these equations. From their reality this general consequence is drawn; viz. when a body, heated in any manner whatever, is placed in a medium the temperature of which is zero, it always attains, before its complete cooling, a regular state in which the temperatures of all its points decrease in the same geometrical progression for equal increments of time. We shall demonstrate in another chapter, that, if that body is a homogeneous sphere, these temperatures will be equal for all the points at an equal distance from the centre, and the same as if the initial heat of each of its concentric strata had been uniformly distributed throughout its extent.
The equations of partial differences upon which depend the laws of cooling in bodies are of the first order in regard to time, whilst the equations relative to the vibrations of elastic bodies and of fluids are of the second order; there result essential differences between the expressions of the temperatures and those of the velocities at a given instant, and for that reason it appears at least very difficult to conceive that the phænomena which may result from a molecular radiation should be equally explicable by attributing them to the vibrations of an elastic fluid. When we have obtained the expressions of the unknown quantities in functions of the time, in either of these kinds of questions, if we make the time in them equal to zero, we deduce from that, series of different forms which represent, for all the points of the system which we consider, arbitrary functions, continuous or discontinuous, of their coordinates. These expressions in series, although we might not be able to verify them, except in particular examples, ought always to be admitted as a necessary consequence of the solution of every problem, the generality of which has been demonstrated à priori; it will however be desirable that we should also obtain them in a more direct manner; and we might perhaps so attain them, by means of the analysis of which I had made use in my first Memoir on the theory of heat, to determine the law of temperatures in a bar of a given length, according to the integral under a finite form of the equation of partial differences.
