192 J. TF. Gibhs — EquiUbrmm of Heterogeneous Substances. 



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 diifers 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 deter- 

 mined by the values of dij, dv, dtn^, dm 2, . . . dw„, it is evident 

 that the change in question Avill caiise the mass to cease to be homo- 

 geneous whenever the expression 



^f '""- % *+ '-i^' '"'•' ••■+^17 *"" <^'''> 



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

 it would become imstable, as Ji„+i would become negative. Hence, 

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

 giving to dr/, dv, dm^, dm^, . . . dm„ the same values taken nega- 

 tively), will cause the mass to cease to be homogeneous. The condi- 

 tion which must be satisfied with refei'ence to dij, dv, diit^, dm^, 

 . . . dm„, in order that neither the change indicated, nor the 

 reverse, shall destroy the homogeneity of the mass, is expressed by 

 equating the above expression 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 vi-f 1 of the differentials dt, dp, f^/',, djx^, . . . dj.i„, 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, /<,, //g, • • . yw„ will be independent. There- 

 fore, 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 u-\-l of the above differentials 

 there will correspond three different phases, of which one is stable 

 with respect both to continuous and to discontinuous changes, another 

 is stable with respect to the former and unstable with respect to the 

 latter, and the third is unstable with respect to both. 



In general, however, if 91 of the quantities p, t, /a ^, /<^, . . . //„, 

 or n arbitrary functions of these quantities, have the same constant 

 values as at a critical phase, the linear series of phases thus deter- 

 mined will be stable, in the vicinity of the critical phase. But if less 



