ART. F 



222 MILNE 



Choose the second process such that A'p = — Ap, thus restor- 

 ing the initial pressure. Then 



Ax + A'x < vAp — V d'p 



J pa — Ap 



But 



Hence 





< 0. 



[(A + A') xl,. V < 0. 



This contradicts (8), and so the imposition of (8) must imply 

 the truth of (9). The proof of the converse may be left to the 

 reader. 



As an example of the application of this criterion we shall 

 prove that Cp, the specific heat at constant pressure, must be 

 positive. Divide a homogeneous specimen of the body into two 

 equal parts, at the same pressure, and take a varied state of 

 the same total entropy in which one part has been heated at 

 constant pressure and the other cooled. Then by the properties 

 of the heat function x already established, we must have, if x 

 refers to unit mass, 



X(77 + Ar?, p) + x{-n - At/, p) > 0, 



since the gain of entropy of the one portion must be equal to 

 the loss of entropy of the other. 



It follows, by expansion by Taylor's theorem, that 



> 0. 



' p 



But since 



/a!x\ 



dx = d{€ + pv) 



