50 THEORY OF HEAT. [CHAP. I. 



Thus the law of the temperatures is expressed for the first body 



by the equation v = a H z, and for the second, by the equa 

 te 



H a 



tion u = a H -, z, v in the first solid, and u in the second, being 



& 



the temperature of the section whose height is z. 



This arranged, we will compare the quantity of heat which, 

 during the unit of time traverses a unit of area taken on an 

 intermediate section L of the first solid, with that which during 

 the same time traverses an equal area taken on the section L 

 of the second, e being the height common to the two sections, 

 that is to say, the distance of each of them from their own 

 base. We shall consider two very near points n and ri in the 

 first body, one of which n is below the plane L and the other 

 ri above this plane : x, y, z are the co-ordinates of n : and x f , y , z 

 the co-ordinates of ri, e being less than z, and greater than z. 



We shall consider also in the second solid the instantaneous 

 action of two points p and p, which are situated, with respect 

 to the section U, in the same manner as the points n and ri with 

 respect to the section L of the firsfc solid. Thus the same co 

 ordinates x, y, z, and of, y , z referred to three rectangular axes 

 in the second body, will fix also the position of the points p 

 and p . 



Now, the distance from the point n to the point ri is equal 

 to the distance from the point p to the point p , and since the 

 two bodies are formed of the same substance, we conclude, ac 

 cording to the principle of the communication of heat, that the 

 action of n on ri, or the quantity of heat given by n to ri, and 

 the action of p on p , are to each other in the same ratio as the 

 differences of the temperature v v and u u. 



Substituting v and then v in the equation which belongs to 



the first solid, and subtracting, we findv v = (z /) ; we 



6 



have also by means of the second equation u u= -, (z z }, 



6 



whence the ratio of the two actions in question is that of to 



a -V 



e 



