96 THEORY OF HEAT. [CHAP. II. 



whose radius is x, and whose length is supposed to be equal 

 to unity, is expressed by 



dx 



To find the quantity of heat which, crossing the second surface 

 whose radius is x + dx, passes from the infinitely thin shell into 

 the part of the solid which envelops it, we must, in the foregoing 

 expression, change x into x + dx, or, which is the same thing, 

 add to the term 



2K7TX ys- dt, 



dx 



the differential of this term, taken with respect to x. Hence 

 the difference of the heat received and the heat lost, or the 

 quantity of heat which accumulating in the infinitely thin shell 

 determines the changes of temperature, is the same differential 

 taken with the opposite sign, or 



*&..*(.*); 



on the other hand, the volume of this intervening shell is Qirxdx, 

 and ZCDjrxdx expresses the quantity of heat required to raise 

 it from the temperature to the temperature 1, C being the 

 specific heat, and D the density. Hence the quotient 



~ 



dx 



ZCDwxdx 



is the increment which the temperature receives during the 

 instant dt. Whence we obtain the equation 



k - K (^ ld JL\ * T ! 



dt CD \da? x dx) \ 



120. The quantity of heat which, during the instant dt, 

 crosses the cylindrical surface whose radius is x t being expressed 



in general by 2Kirx -j- dt, we shall find that quantity which 



escapes during the same time from the surface of the solid, by 

 making x = X in the foregoing value; on the other hand, the 



