328 THEORY OF HEAT. [CHAP. VIII. 



rature at every point decreases proportionally to the powers of 

 the fraction e~ skn ^- } the successive states are then all similar, or 

 rather they differ only in the magnitudes of the temperatures 

 which all diminish as the terms of a geometrical progression, pre 

 serving their ratios. We may easily find, by means of the pre 

 ceding equation, the law by which the temperatures decrease from 

 one point to another in direction of the diagonals or the edges of 

 the cube, or lastly of a line given in position. We might ascer 

 tain also what is the nature of the surfaces which determine the 

 layers of the same temperature. We see that in the final and 

 regular state which we are here considering, points of the same 

 layer preserve always equal temperatures, which would not hold 

 in the initial state and in those which immediately follow it. 

 During the infinite continuance of the ultimate state the mass is 

 divided into an infinity of layers all of whose points have a com 

 mon temperature. 



339. It is easy to determine for a given instant the mean 

 temperature of the mass, that is to say, that which is obtained by 

 taking the sum of the products of the volume of each molecule 

 by its temperature, and dividing this sum by the whole volume. 



We thus form the expression 1 1 1 3 % , which is that of the 



mean temperature V. The integral must be taken successively 

 with respect to x, y, and z, between the limits a and a : v being 

 equal to the product X YZ } we have 



thus the mean temperature is fl-gpl &amp;gt; s i nce the three complete 

 integrals have a common value, hence 



e-^+ Ac. 



V nfl J PI \ n t a 



The quantity na is equal to e, a root of the equation e tan e = -~ , 



and //, is equal to x (l + 5 J We have then, denoting the 

 different roots of this equation by 6 1} e a , e 8 , &c., 



