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, 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 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 phzeno- 
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 points 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 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. 
Vou. I.—Parr I. K 
