EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 137 



A 



-, the value of A will be positive, and will be independent of m r 



m i m 



Then for small values of - ? we have by (210), approximately, 



7/C/l 



(213) 



If we write the integral of this equation in the form 



(215) 



like A will have a positive value depending only upon the tem- 

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

 in which the value of m 2 is very small compared with m 1? we may 



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



and write (7m 



*> (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 proportion of two components. 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 

 quantities A and C 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 $ 2 the body is composed. If 

 the quantities of any of the components besides $ 2 are very small 

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

 that the value of /z 2 , and therefore the values of A and (7, will be 

 nearly the same as if these components were absent. 



Hence, if the independently variable components of any body are 

 S a , ...Sy, and S h ,...S k , the quantities of the latter being very small 

 as compared with the quantities of the former, and are incapable of 

 negative values, we may express approximately the values of the 

 potentials for S h ,...S k by equations (subject of course to the uncer- 

 tainties of the assumptions which have been made) of the form 



, (217) 



, (218) 



in which A h , C h , ... A k , C k denote functions of the temperature, the 

 pressure, and the ratios of the quantities m a , ... m a . 



