216 MILNE ART. F 



Now choose A't = —At, thus restoring the initial temperature 

 (a state for which \l/ is defined is of course necessarily a state 

 of uniform temperature). We have then 



At/' + AV < - riM + i^d't, 



where now to denotes the initial temperature. This gives 



At/' + AV < - -^0 Ai + / ° Uo + f-^l (t-to) + .. .\d% 



where t/o denotes the initial entropy. Evaluating the integral 

 we have 



At^ + AV < - h(jX ^^^^' + • • • 



Now ( — ) is positive. Hence, provided A^ is sufficiently small, 

 \dt/a 



Ai/- + AV < 0. 



We have thus constructed a state for which the total (finite) 

 increment in ^, namely (A + A')\l/, is negative, contradicting 

 (1). Moreover it is a state of the same (initial) temperature 

 and volume. This demonstrates that (1) implies (2). The proof 

 of the converse may be left to the reader. The above estab- 

 lishes for a finite change Gibbs' result [HI], established by him 

 by less rigorous methods in equations [112] and [115] (Gibbs, 

 I, 91). 



S. The Free Energy Function. In equation [117] Gibbs states 

 without proof that the condition of equilibrium may be written 



We shall prove that 



and 



are equivalent. 



(A1A)^« ^0 (5) 



(Ar)^p^O (6) 



