176 BUTLER ART. D 



Now, we may express dfj,i as a function of p, t, mi by the equa- 

 tion 



This equation may be applied to either of the two phases. 

 Applying it to the first phase, we may write, by (158) and (159), 



\dp Jt.m ' ' \dt /p. m 



Hence, substituting in (273) the value of d^ given by these 

 equations and rearranging, we find 



{{v" - v') - (mi" - miO vA dp 

 = [(V - r?') - (mi" - m/) ^i'] dt 



+ (mi" - miO ( ^Y ' dmi'. (275) 



Similarly, when the terms of (274) are determined by the 

 second phase, we obtain 



[{v" - v') - (mi" - miO vi"\ dp 

 - Kv" -v) - (wi" - miO vi"] dt 



+ (mi" - miO (j^Y • dmi". (276) 



\dmi/p, I, mj 



In order to interpret these equations we may first observe that 

 v' is the volume of the quantity of the first phase which contains 

 mi' of the first component. Thus [v' + (m/' — m/) {dv'/dmi')] 

 is approximately equal to the volume of that quantity of 

 this phase which contains m/' of this substance. Hence we 

 see that [v" — v' — (m/' — m/) y/] is approximately equal 

 to the difference of the volumes of quantities of the two phases 

 containing the same amount (viz., m/') of this substance. In 

 the same way [v" — v' — (m/' — mi)vi"] is the approximate 



