220 MILNE ART. F 



and accordingly 



/•po + A'p 



Af + AY <vL-p + \ V d'p. 



J po 



Now choose A'p = —Ap, thus restoring the initial pressure. 

 Then 



Ar + A'r < .oAp - lljn + (|)/p - .0) + ...] d'p 



Now I 7- ) is negative. Hence 

 \dp/o 



[(A + A')r]^p <0. 



4. The Heat Function. We shall now prove that the varia- 

 tional conditions 



(Ax),.p^O (8) 



and 



(Ae),.„^0 (9) 



are equivalent. These criteria are not stated by Gibbs, but 

 clearly there must be a parallel set of criteria involving the 

 heat function. 



To prove that (8) implies (9) let us suppose there is a 

 neighbouring state for which 



(A€)„„ < 0. 



We shall prove that this implies the existence of a neighbouring 

 state violating (8). Hence if we know that (8) holds, (9) must 

 also hold. 



If this neighboring state is not one of uniform pressure, let 

 the pressure equalize itself. This can only increase the entropy, 

 and thus we have a state of the same energy and volume, and 

 greater entropy. Now remove heat at constant volume until 



