SURFACES OF DISCONTINUITY 611 



the system is definitely unstable in the first state. However, 

 it may be possible that the expression (25) is positive for 

 infinitesimally small values of a" — a', iij' — nj, nh" — iJ-h, etc., 

 but would be negative for finite values of these changes. The 

 system would satisfy the theoretical conditions of stability 

 which, as any student of dynamics knows, only compare the 

 state of a system with other states infinitesimally near it. Yet 

 the system, as Gibbs points out, would not be stable in the 

 practical sense; for a disturbance which, while being small, 

 would be sufficient to carry the system beyond the infinitesi- 

 mally near states of larger energy would bring it to states of less 

 energy from which it would not tend to return to the first state. 

 Perhaps it may not be out of place to remind the reader that the 

 quantities Dnia^', Drria^", etc., are not variations of mass; they 

 are the small masses initially and finally present in the small 

 part of the film. Further, that Drria^" — Drtia^', etc., are not 

 necessarily small compared to Dma^', Drria^", etc. They are 

 small compared to the masses in the rest of the film and the 

 homogeneous masses; that is why we can use them correctly in 

 the expression (24). But since they are finite changes in 

 respect to the small portion Ds of the system, they produce 

 finite changes in the surface tension and the g,h, ... potentials 

 there, so that we can regard a" — o-', iig" — Hg, nh" — fj-h, etc., 

 as finite differences if necessary. This small digression on the 

 meaning of the D symbol may serve to illuminate the point 

 about practical instability. 



The argument can now be extended to the whole film. 

 Having effected the change in one small part of the film, we can 

 carry it out for another small part, changing entropy and masses 

 there so as to produce the g,h, ... potentials and surface tension, 

 fjLg", iJLh", . . . cr", which exist in the first small part, and so on. 

 This is simply the procedure indicated by the integrations on 

 Gibbs, I, 240. The changed condition in the film is therefore 

 uniform in nature throughout and is one which could exist in 

 equilibrium with the homogeneous masses in their practically 

 unchanged condition. The difference of energy in the whole 

 system for the two states of the film is 



(a" - a')s + w/'(m/' - m/) + mtS"{y^H" - n[) + . . . (26) 



