M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 129 



which I have since made a great number of applications, and which 

 I shall also constantly follow in this work. The Sixth Chapter con- 

 tains already the application to the general equations of the mo- 

 tion 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, inde- 

 pendently 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 me- 

 dium the temperature of which is zero, it always attains, befoi'e 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 equa- 

 tions 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 phaeno- 

 mena which may result from a molecular radiation should be equally ex- 

 plicable 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 jioints of the system which we consider, arbitrary 

 functions, continuous or discontinuous, of their coordinates. These ex- 

 pressions in series, although we might not be able to verify them, except 

 in particular examples, ought always to be admitted as a necessary con- 

 sequence of the solution of every problem, the generality of which has 

 been demonstrated a priori ; it will however be desirable that we should 

 also obtain them in a more direct manner ; and we might perhaps so at- 

 tain them, by means of the analysis of which I had made use in my first 

 Memoir on the theory of heat, to determuie 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. 



Vol. I. — Pakt I. k 



