110 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



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

 if we regard t, v, m 2 , . . . m n as having the constant values indicated 

 by accenting these letters, yu x shall be an increasing function of m lr 

 when the variable phase differs 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 x (within these limits) for any 

 constant values of t, v, m 2 , . . . m n . That is, 



>0. (167) 



t f Vj Wl2) ... ^ 



When this condition is satisfied, as well as (166), $ will have a 

 minimum value, for any constant values of v, m 2 , . . . m n , when t t f 

 and /*! = //!'; so that in applying the general condition of stability 

 we need only consider the phases for which t = t' and JJL^ = ///. 



In this way we may also obtain the following particular conditions 

 of stability : 



>0> (168 > 



m n 



><>. (169) 



^, ...,,_, 



When the n + 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 r\, m t , . . . m n 

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

 t, fji v ... fJL n . In this way we may obtain different sets of n+1 

 conditions which are necessary and sufficient for stability. The 

 quantity v might be included in the first of these lists, and p in 

 the second, except in cases when, in some of the phases considered, 

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

 zero. Then the condition that that quantity shall be constant would 

 create a restriction upon the variations of the phase, and cannot be 

 substituted for the condition that the volume shall be constant in 

 the statement of the general condition of stability relative to the 

 minimum value of $. 



To indicate more distinctly all these particular conditions at once, 

 we observe that the condition (144), and therefore also the condition 

 obtained by interchanging the single and double accents, must hold 



