EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 133 



and V for the determinant formed from this by substituting for the 

 constituents in any horizontal line the expressions 



dU_ dU_ dU 



dm^ dm 2 ' cra n _i' 



the equations of critical phases will be 



tf=0, F=0. (208) 



It results immediately from the definition of a critical phase, that 

 an infinitesimal change in the condition of a mass in such a phase 

 may cause the mass, if it remains in a state of dissipated energy (i.e., 

 in a state in which the dissipation of energy, by internal processes is 

 complete), to cease to be homogeneous. In this respect a critical phase 

 resembles any phase which has a coexistent phase, but differs from 

 such phases in that the two parts into which the mass divides when 

 it ceases to be homogeneous differ infinitely little from each other and 

 from the original phase, and that neither of these parts is in general" 

 infinitely small. If we consider a change in the mass to be determined 

 by the values of drj, dv, d^, dm z ,...dm n , it is evident that the 

 change in question will cause the mass to cease to be homogeneous 

 whenever the expression 



has a negative value. For if the mass should remain homogeneous, 

 it would become unstable, as M n +i would become negative. Hence, in 

 general, any change thus determined, or its reverse (determined by 

 giving to drj, dv, dm 1} dm 2 , ... dm n the same values taken negatively) 

 will cause the mass to cease to be homogeneous. The condition which 

 must be satisfied with reference to drj, dv, dm l} dm 2 , ... dm n , in order 

 that neither the change indicated, nor the reverse, shall destroy the 

 homogeneity of the mass, is expressed by equating the above expres- 

 sion to zero. 



But if we consider the change in the state of the mass (supposed to 

 remain in a state of dissipated energy) to be determined by arbitrary 

 values of n + l of the differentials dt, dp, dju v d/ui 2 , ... dfi n , the case 

 will be entirely different. For, if the mass ceases to be homogeneous, 

 it will consist of two coexistent phases, and as applied to these, only n 

 of the quantities t, p, fjL l} jUL z ,...ju. n will be independent. Therefore, 

 for arbitrary variations of n+l of these quantities, the mass must in 

 general remain homogeneous. 



But if, instead of supposing the mass to remain in a state of dissi- 

 pated energy, we suppose that it remains homogeneous, it may easily 

 be shown that to certain values of n+l of the above differentials 



