612 RICE ART. L 



41. Three Conclusions Drawn from the Analysis in Subsection (40) 



This disposes of the analytical steps on these pages of Gibbs' 

 treatise. There are three conclusions based on them. The first 

 appears at the top of page 240. As presented it is somewhat 

 elusive, but we can put it as follows. It is possible that the 

 potentials n/\ iih", . . . which correspond to the masses Wa^", 

 nih^", . . . mg^", ruh^", . . . may be respectively equal to the 

 potentials ix/, nh, . . . which correspond to Ma^', nih^', . . . 

 fn/', nih^', . . . (Of course, the potentials ^a, M6, • • • remain 

 unchanged in any case.) If this is so, then by (26) {a" — (t')s 

 must be positive if the single accent state is to be a stable state 

 of equilibrium; i.e., g" > a'. There appears to be a contradic- 

 tion here; we have seen that o- is a function of t and the potentials 

 Mo^, Mb^> • ■ • M(7^, fJ'h^, • ■ ■ and it appears absurd to assume that 

 <t" is different from o-' at all if Ha, Hb, ■ . ■ \i.g\ t^h, ... do not 

 differ in value from Ha, M6> • • • )"»", M^", • • • But this is to over- 

 look the possibility of a being a double-valued or multi-valued 

 function of the temperature and potentials, so that if the 

 variables ^a, M6> • • • M^j y-n, . ■ • experience a change of values 

 corresponding to changes in the masses of the components, and 

 presently retake the same values, the surface tension may not 

 retake its original value. (We have already made use of this 

 result in an earlier part of this commentary to show that if 

 there are, say, a "gaseous" and a "liquid" phase in the surface 

 of discontinuity, they must, if stable, have the same value of a.) 



The second conclusion drawn concerns the sign of a. In 

 the argument so far there has been no displacement or def- 

 ormation of Ds. It is implied also that s is practically plane. 

 If Ds being plane is deformed, its area must increase. This 

 will necessitate the withdrawal of small amounts of the com- 

 ponents from the homogeneous masses or from the rest of the 

 film in order to maintain the nature of the film in Ds unchanged. 

 These amounts, as before, will be infinitesimal for the rest of the 

 system. The amounts will have gone from a place where the 

 potentials have been at certain values to a place where they are 

 at the same values. This will cause no change in the energy 

 of the system; the term of the energy expression which will 

 have altered will be aDs which will become a{Ds + 8Ds). 



