SURFACES OF DISCONTINUITY 613 



The energy change will be adDs. For stability this must be 

 positive, and as 8Ds is positive, a must be positive. The 

 paragraphs on pages 240, 241 elaborate this. 



The third conclusion occurs in the paragraph beginning 

 towards the bottom of Gibbs, I, 241. It is very elusive indeed 

 and the final sentences of the paragraph are not very happily 

 chosen for a reader not expert in mathematical technique. First 

 of all the reader must realize that there may be a whole con- 

 tinuous series of states of the system differing in the nature of 

 the film, which will be states of stable equilibrium. A change 

 from any one of them to any state infinitesimally near it, whether 

 a non-equilibrium state or one of its equilibrium neighbors, will 

 involve an increase of energy. Let the single and double accents 

 refer to two neighboring infinitesimally different states of stable 

 equilibrium. We have seen then that 



(a" - a')s + W - m/)w/" + W - Hk')mH'" + • • • 



must be positive. But exactly the same reasoning will show 

 that 



{a' - a")s + (m/ - lij')^/ + U' - tJ^h")m,^' + • • . 



must also be positive. Now write fXg for /x/', Hh for ixh^', . . . 

 Hg -{■ Afig for fXgS", fx h ^ A)U/, for nh^", etc.; o- for a', (t -\- Acr 

 for a", m/ for Mg^', Mh^ for Mh^' , ... m^ + Anig for m/", 

 rrih + Anih for w^-s", etc. From the expression given four lines 

 above we obtain the result 



s(-Ao-) + m/(- Arrig) + mh^{— Amn) + ... > 0, 



which is just the equation preceding [521]. Considering [521] we 

 may write it, remembering that Hg, Hh, . . . are the only quantities 

 which are varying, 



d<r da 



—— Aflg + — AHh + . . . 

 OUg dfXh 



1 r av , , av , , av "1 



+ 2{^' (^''«)' + w ^^'-'^ • • ■ + %7;^ ^"'^"^ + ■■■■} 



-t- higher powers < — TgAfjLg — ThAfXH — . .. 



