196 J. W. Gibbs — Equilibrium of Heterogeneous Substa7ices. 



denoted by *S'i and S^, the relation between //g and ^ tor any con- 

 stant temperature and pressure and for such small values of -^ that 



the differential coefficient in (211) may be regarded as having the same 

 constant value as when m^ = 0, the values of t, p, and m ^ being un- 

 changed. If we denote this value of the differential coefficient by 



— the value of ^ will be positive, and will be independent of m^. 



m^ ' 



Then for small values of '^, we have by (210), approximately, 



^2 



i. e., 



^\dm2/t, p, m, 

 \rtlog rn2/t,p, Ml 



If we write the integral of this equation in the form 



pi2=Alog-^^, (215) 



J^ like A will have a positive value depending only upon the tempera- 

 ture and pressure. As this equation is to be applied only to cases in 

 which the value of m^ is very small compared with ^)t^, we may 



regard — - as constant, when temperature and pressure are constant, 



and write 



p(^ = A\og—^, (216) 



C denoting a positive quantity, dependent only upon the temperature 

 and pressure. 



We have so far considered the composition of the body as varying 

 only in regard to the proj^ortion of two comi^onents. But the argu- 

 ment will be in no respect invalidated, if we suppose the composition 

 of the body to be capable of other variations. In this case, the quan- 

 tities A and 6' will be functions not only of the temperature and 

 pressure but also of the quantities which express the composition of 

 the substance of which together with S^ the body is composed. If 

 the quantities of any of the components besides yS'a are very small 

 (relatively to the quantities of others), it seems reasonable to assume 

 that the value of ju^, and therefore the values of .1 and C, will be 

 nearly the same as if these components were absent. 



