54 THEORY OF HEAT. [CHAP. I. 



the upper plane, instead of being maintained as formerly at a 

 less temperature b, is exposed to atmospheric air maintained 

 at that temperature b, the perpendicular distance of the two 

 planes being denoted always by e : the problem is to determine 

 the final temperatures. 



Assuming that in the initial state of the solid, the common 

 temperature of its molecules is b or less than b, we can readily 

 imagine that the heat which proceeds incessantly from the source 

 A penetrates the mass, and raises more and more the tempera 

 tures of the intermediate sections ; the upper surface is gradually 

 heated, and permits part of the heat which has penetrated the 

 solid to escape into the air. The system of temperatures con 

 tinually approaches a final state which would exist of itself if 

 it were once formed; in this final state, which is that which 

 we are considering, the temperature of the plane B has a fixed 

 but unknown value, which we will denote by ft, and since the 

 lower plane A preserves also a permanent temperature a, the 

 system of temperatures is represented by the general equation 



v = a + - z, v denoting always the fixed temperature of the 



section whose height is z. The quantity of heat which flows 

 during unit of time across a unit of surface taken on any section 



whatever is fr - , % denoting the interior conducibility. 



We must now consider that the upper surface B, whose 

 temperature is ft, permits the escape into the air of a certain 

 quantity of heat which must be exactly equal to that which 

 crosses any section whatever L of the solid. If it were not so, 

 the part of the mass included between this section L and the 

 plane B would not receive a quantity of heat equal to that 

 which it loses; hence it would not maintain its state, which is 

 contrary to hypothesis ; the constant flow at the surface is there 

 fore equal to that which traverses the solid : now, the quantity 

 of heat which escapes, during unit of time, from unit of surface 

 taken on the plane B, is expressed by li(ft-b), b being the 

 fixed temperature of the air, and h the measure of the conduci 

 bility of the surface B\ we must therefore have the equation 



V~T~ = h(@- b), which will determine the value of ft. 



