EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 255 



may regard a- as positive (for if & is not positive when p' =>", 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 with respect to its perimeter), we see by (550) that the 

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

 a; p'p", and r 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, W, r, and a- have single, real, and positive values until 

 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. 225-227), that the globule must become insensible in magni- 

 tude before r can vanish. 



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

 work which would be required 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 [;/], [wj, [m 2 ], 

 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), , r n r n r n 



- 1 M ~ A*i OJ - 02 M - etc - 



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

 the globule is formed. But [e] 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 

 required 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 pressure 

 p" (supposed less than p', the pressure belonging to a different phase 

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

 those limits within which the method used has been justified, the 

 mass in question must be regarded as in strictness stable with respect 

 to the growth of a globule of the kind considered, since W, the work 



