SECT. TIL] MOVEMENT IX THREE DIMENSIONS. 77 



The sum of the two actions of m on fj, and of m on // is there 

 fore 2qcy. 



Suppose then that the plane H belongs to the infinite solid 

 whose temperature equation is v = A + cz, and that we denote 

 also by m t JJL and p those molecules in this solid whose co 

 ordinates are x, y, z for the first, x + a, y + /3, z 4- 7 for the second, 

 and x a,y j3,z+y for the third : we shall have, as in the 

 preceding case, v-w + v-w = - 2cy. Thus the sum of the two 

 actions of m on //- and of m on p, is the same in the infinite solid 

 as in the prism enclosed between the six planes at right angles. 



We should obtain a similar result, if we considered the action 

 of another point n below the plane H on two others v and v , 

 situated at the same height above the plane. Hence, the sum 

 of all the actions of this kind, which are exerted across the plane 

 H, that is to say the whole quantity of heat which, during unit 

 of time, passes to the upper side of this surface, by virtue of the 

 action of very near molecules which it separates, is always the 

 same in both solids. 



95. In the second of these two bodies, that which is bounded 

 by two infinite planes, and whose temperature equation is 

 v = A + cz, we know that the quantity of heat which flows during 

 unit of time across unit of area taken on any horizontal section 

 whatever is cK, c being the coefficient of z, and K the specific 

 conducibility ; hence, the quantity of heat which, in the prism 

 enclosed between six planes at right angles, crosses during unit 

 of time, unit of area taken on any horizontal section whatever, 

 is also - cK y when the linear equation which represents the tem 

 peratures of the prism is 



v = A + ax + by + cz. 



In the same way it may be proved that the quantity of heat 

 which, during unit of time, flows uniformly across unit of area 

 taken on any section whatever perpendicular to x, is expressed 

 by - aK, and that the whole quantity which, during unit of time, 

 crosses unit of area taken on a section perpendicular to y, is 

 expressed by bK. 



The theorems which we have demonstrated in this and the 

 two preceding articles, suppose the direct action of heat in the 



