J. W. Gibbs — Equilibrium of Heterogeneous Stibstances. 419 



we may regard G as positive (for if G is not positive when p'^=Lp\ the 

 surface when plane would not be stable in regard to position, as 

 it certainly is, in every actual case, when the proper conditions are 

 fulfilled Avith respect to its perimeter), we see by (550) that the pres- 

 sure in the interior mass must be the greater; i. e., we may regard 

 o", p' —p>\ ^iitl *' as all positive. By (555), the value of W will 

 also be positive. But it is evident from equation (552), which defines 

 W. that the value of this quantity is necessarily real, in any possible 

 case of equilibrium, and can only become infinite when r becomes 

 infinite and p'^p". Hence, by (556) and (558), as p' —p" increases 

 from very small values, TP^ r, and a have single, real, and positive 

 values xmtil they simultaneously reach the value zero. Within this 

 limit, our method is evidently applicable ; beyond this limit, if 

 such exist, it will hardly be profitable to seek to interpret the 

 equations. But it must be remembered that the vanishing of the 

 radius of the somewhat arbitrarily determined dividing surface may 

 not necessarily involve the vanishing of the physical heterogeneity. 

 It is evident, however, (see pp. 387-389,) that the globule must be- 

 come insensible in magnitude before /• can vanish. 



It may easily be shown that the quantity denoted by W is the 

 work which would be requii-ed to form (by a reversible process) the 

 heterogeneous globule in the interior of a very large mass having 

 initially the uniform phase of the exterior mass. For this work is 

 equal to the increment of energy of the system when the globule is 

 formed without change of the entropy or volume of the whole system 

 or of the quantities of the several components. Now [//], [«<,], [''^o], 

 etc. denote the increments of entropy and of the components in the 

 space where the globule is formed. Hence these quantities with the 

 negative sign will be equal to the increments of entropy and of the 

 components in the rest of the system. And hence, by equation (86), 



will denote the increment of energy in all the system except where 

 the globule is formed. But [f] denotes the increment of energy in 

 that part of the system. Therefore, by (552), W denotes the total 

 increment of energy in the circumstances supposed, or the work re- 

 quired for the formation of the globule. 



The conclusions which may be drawn from these considerations 

 with respect to the stability of the homogeneous mass of the pres- 

 sure p" (supposed less than ^>', the pressure belonging to a different 

 phase of the same temperature and potentials) are very obvious. 



