212 fT. W. Gibhs — Equilibrium of Heterogeneous Substances. 

 whence, by the general equation (88), 



1] = m { H+ c log «; + a log — J , (262) 



am t , . 



/> = — - (263) 



c + a - Il—c\ogt-\-alog — \. (264) 



From (260), by (87) and (91), we obtain 



'C, = Em, -\- ')nt\c — H — c log t + a log — ] + p v, 



and eliminating v by means of (263), we obtain the fundamental equa- 

 tion 



? = Eyn + m tic + a - H - {c-^a) log ^ + a log — |. (265) 



From this, by differentiation and comparison with (92), we may 

 obtain the equations 



// z=. m (Hi- (c + a) log « — a log — |, (266) 



a m t 



^=-^, (267) 



lx = E -{- tic + a — H - (e+«) log t + a log — j. (268) 



The last is also a fundamental equation. It may be written in the 

 form 



or, if we denote by e the base of the Naperian system of logarithms, 



E—c—a c + a fi—E 

 p = ae "■ t "" e ""^ (270) 



The fundamental equation between Xi V-, Pi ^"d m may also be 

 easily obtained ; it is 



(c+«)log7 =--H+a\og^, (271) 



^ ^ * {c-\-a)m m ^ a' ^ ' 



which can be solved with respect to x- 



Any one of the fundamental equations (255), (260), (265), (270), 

 and (271), which are entirely equivalent to one another, may be 



