SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 105 



which the point M divides into two equal parts ; denote by 

 x, y, z the co-ordinates of the point M t and its temperature by 

 v, the co-ordinates of the point p by x + a, y + /3, z + y, and its 

 temperature by w, the co-ordinates of the point m by as a, y fi, 

 z y, and its temperature by u t we shall have 



v = A ax ly cz, 



whence we conclude that, 



v w = az + 6/3 + cy, and u v = az + b/3 + cy ; 

 therefore v w = u v. 



Now the quantity of heat which one point receives from 

 another depends on the distance between the two points and 

 on the difference of their temperatures. Hence the action of 

 the point M on the point //, is equal to the action of m on M; 

 thus the point M receives as much heat from m as it gives up 

 to the point p. 



We obtain the same result, whatever be the direction and 

 magnitude of the line which passes through the point J/, and 

 is divided into two equal parts. Hence it is impossible for this 

 point to change its temperature, for it receives from all parts 

 as much heat as it gives up. 



The same reasoning applies to all other points ; hence no 

 change can happen in the state of the solid. 



133. COROLLARY I. A solid being enclosed between two 

 infinite parallel planes A and B, if the actual temperature of 

 its different points is supposed to be expressed by the equation 

 v = lz, and the two planes which bound it are maintained 

 by any cause whatever, A at the temperature 1, and B at the 

 temperature ; this particular case will then be included in 

 the preceding lemma, if we make A=l, a = 0, & = 0, c = 1. 



134. COROLLARY II. If in the interior of the same solid 

 we imagine a plane M parallel to those which bound it, we see 

 that a certain quantity of heat flows across this plane during 

 unit of time ; for two very near points, such as m and n, one 



