214 MILNE ART. F 



but there exist variations for which 



(At;). > Oor(A€), < 0. 



In the above, the subscript denotes that the corresponding 

 variable is maintained constant in the variation. 



Gibbs proceeds, in the section under consideration (Gibbs, I, 

 89-92), to estabhsh the equivalence of the above to similar 

 variational conditions involving 



(1) the work function yp, defined hy ^p = e — t-q, 



(2) the heat function x, defined by x = « + P^, 



(3) the free energy function f , defined hy ^ = e — tr] -\- pv. 

 He gives a method of proof which is sound in principle, and 

 which suggests the method to adopt, but which does not dis- 

 tinguish between small variations and finite variations. The 

 following includes the substance of Gibbs' results, and supplies 

 proofs in certain cases where Gibbs left the proof to the reader. 



2. The Work Function. The value of the criteria about to be 

 discussed is that they render the general criteria more easily 

 applicable to certain particular cases, by restricting the type 

 of variation permitted. For example, in certain cases they 

 impose a condition of constancy of volume in addition to 

 constancy of entropy, in discussing changes of energy. 



We shall now prove that the condition 



W),.v^O (1) 



is equivalent to the condition 



(A6),.„^0. (2) 



For suppose that there exists a neighbouring state for which 



(Ae),., <0. 

 We shall prove that there then exists a state for which 



(A^),,„ < 0. 



This will ensure that if we are given that (1) is true, no con- 

 tradiction of (2) can exist; hence (1) implies (2). 

 For, if the neighbouring state for which (Ae),, , < is not 



