J. W. Gibbs — Equilibrium of Heterogeneous Substances. 167 



When this condition is satisfied, the value of ^, foi* any ijiven vahies 

 oft?, wij, . . . ;/^„ will be a minimum when t-=.t'. And therefore, in 

 applying the general condition of stability relating to the value of 

 <^, we need only consider the phases for which t = t'. 



We see again by (165) that the general condition requires that 

 if we regard ^, y, ni^., . . . m„ as having the constant values indicated 

 by accenting these letters, //j shall be an increasing function of m,, 

 when the variable phase difters sufficiently little from the fixed. But 

 as the fixed phase may be any one within the limits of stability, /.i , 

 must be an increasing function of m j (within these limits) for any 

 constant values of ^, v, mg, . . . m„. That is, 



(i^) >0- (16V) 



When this condition is satisfied, as well as (166), ^ will have a min- 

 imum value, for any constant values of v, m^^ . • . ?/*„, when t=it' 

 and yu, = ///; so that in applying the general condition of stability 

 we need only consider the phases for which t-=.t' and //j = yu/. 



In this way we may also obtain the follov\^ing particular conditions 

 of stability : 



(4^) >0, (168) 



\nm^lt^ w, m,, ma, . . . ??i„ 



(4^\ >0. (169) 



\Amjt, V, mi, . . . m„_, 



When the 7i-\- 1 conditions (166)-(169) are all satisfied, the value 

 of ^, for any constant value of v, will be a minimum when the tem- 

 perature and the potentials of the variable phase are equal to those 

 of the fixed. The pressures will then also be equal and the phases 

 will be entirely identical. Hence, the general condition of stability 

 will be completely satisfied, when the above particular conditions are 

 satisfied. 



From the manner in which these particular conditions have been 

 derived, it is evident that we may interchange in them a/, m^, . . . m„ 

 in any way, provided that we also interchange in the same way 

 ^, //,, . . . //„. In this way we may obtain different sets of n -\- 1 

 conditions which are necessary and sufficient for stability. The quan- 

 tity V might be included in the first of these lists, and ~ p in the 

 second, except in cases w^hen, in some of the phases considered, the 

 entropy or the quantity of one of the components has the value zero. 



