106 J. W. Gihhs — Equillhrhim of Heterogeneous Substances. 



■c^" + t" if -p" v" - fx.'m," . . . - //„' m„" 

 -t,' -t'lf ->rp'v" +fi,'m^' . . . +//„'m„'>0. (161) 



Therefore, for the stability of all phases within any given limits it is 

 necessary and sufficient that within the same limits 



[JC + //^« - v44„<0, (162) 



and 



[A^- fx^Am^ . . . -<-yW„Jm„],,>0, (.163) 



as may easily be proved by the method used with (153) and (154). 

 The first of these formulae expresses the thermal and mechanical con- 

 ditions of stability for a body considered as michangeable in compo- 

 sition, and the second the conditions of chemical stability for a body 

 considered as maintained at a constant temperature and pressure. If 

 '/i= 1, the second condition falls away, and as in this case ? = m/<, 

 condition (162) becomes identical with (148). 



The foregoing discussion will serve to illustrate the relation of the 

 general condition of stability in regard to continuous changes to 

 some of the principal forms of fundamental equations. It is evident 

 that each of the conditions (146), (149), (154), (162), (163) involve 

 in general several particular conditions of stability. We will now 

 give our attention to the latter. Let 



fp ■= € — t' 7/ +p' V — ^i^' )n^ . . . — /<„'«<„, (164) 



the accented letters referring to one phase and the unaccented to 

 another. It is by (142) the necessary and sufficient condition of the 

 stability of the first phase that, for constant values of the quantities 

 relatino- to that phase and of v, the value of $ shall be a minimiim 

 when the second phase is identical with the first. Diflerentiating 

 (164), we have by (86) 



d^ = {t - t') ch] — {p —jo') dn + (//j — /i/) dm^ 



... - (Af„ - /^„')f?m„. (165) 



Therefore, the above condition requires that if we regard v,m^, . . . 

 m„ as having the constant values indicated by accenting these letters, 

 t shall be an increasing function of ;/, when the variable phase differs 

 sufficiently little from the fixed. But as the fixed phase may be any 

 one within the limits of stability, t must be an increasing function of 

 // (within these limits) for any constant values of v, 'm^, . . . m,^. 

 This condition may be written 



(j4J ^^- (^^^) 



X^ijlv, nit, . . . m„ 



