X, ^, r, AND THE CRITERIA OF EQUILIBRIUM 215 



one of uniform temperature, let its temperatures be equalized 

 at constant volume. This can only increase its entropy. Now 

 remove heat so as to reduce the entropy to the initial value, at 

 the same volume. This process reduces the energy. Thus we 

 have constructed a state of uniform temperature for which 



(Ae),,„ < 0. 

 Now we have \p = ^ — tv, whence in general 



ArJ/ = Ae — tAr] — rjAt — AtArj. 



In our case 



At; = 0, and so A\f/ = Ae — r]At 

 or 



A^p + v^t = Ae < 0, (3) 



by hypothesis. 



Now add or subtract heat at constant volume. For such a 

 process the infinitesimal increment in energy, say rf'c is given by 



d'e = t d'-n, 

 whilst similarly 



d'\p = d't - -nd't - t d'-n, 

 i.e., 



d'^ = -r,d't. 

 It follows that the fi7nte increment in \l/, namely A'\p, is given by 



/t+A't 

 r, d't. (4) 



Accordingly, by (3) and (4), 



A\P + AV < - 7?Af + jv d't. 



J t + A't 



