EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 183 



From this equation, by differentiation and comparison with (98), we 

 obtain /*-. 



v 



(344) 



ft-B l 3 n-S-t 



*6**. (345) 



From the general equation (93) with the preceding equations the 

 following is easily obtained, 



i e ait +L 2 (c 2 t+E 2 )t a *e **' . (346) 



v 



We may obtain the relation between p, t, v, and m by eliminating 

 fi from (342) and (345). For this purpose we may proceed as follows. 

 From (342) and (345) we obtain 



(347) 



* * (348) 



and from these equations we obtain 





- 2 * - 2 log 01 -p = (i - a,) log (a! - a 2 ) 



rr _ 



-I- aj log Zj - a 2 log Z/ 2 + (A - c 2 + aj a 2 ) log * - - ^ -. (349) 



(In the particular case when a x = 2a 2 this equation will be equivalent 

 to (333).) By (347) and (348) we may easily eliminate JUL from (346). 



The reader will observe that the relations thus deduced from the 

 fundamental equation (342) without any reference to the different 

 components of the gaseous mass are equivalent to those which relate 

 to the phases of dissipated energy of a binary gas-mixture with 

 components which are equivalent in substance but not convertible, 

 except that the equations derived from (342) do not give the quantities 

 of the proximate components, but relate solely to those properties 

 which are capable of direct experimental verification without the aid 

 of any theory of the constitution of the gaseous mass. 



The practical application of these equations is rendered more simple 

 by the fact that the ratio 04 : a 2 will always bear a simple relation to 

 unity. When a^ and a 2 are equal, if we write a for their common 

 value, we shall have by (342) and (345) 



pv = ami, (350) 



