Pressures in Liquid Mixtures. 103 



nor infinite for any values of the afs. Regarding the p'$ as 

 functions of all the a? s, put 



In 



(50 = Ws (5 = x> 2 ' 3 ' ■ * •' ^ ; ■ • (2i) 



then u t is, by condition c, a continuous singly-valued function 

 of the ay's, finite for all values of the .r's. From (21) 



follows * 



?. = *:*«*' («- 1,2,3,...,*), . . . (22) 



where, by (17), 



u, = In P s for x, = 1 (« = 1, 2, 3, . . ., *). . (23) 



Sow let a? K , the p's, and the w's be expressed in terms of 

 #i, «r 2 , t r 3 , . . . , « K _i. Then, for 



a? g = (s = 1, 2, 3, . . . , «— 1), 



we have, by definition of the derivative, (22), and (15), that 



-^* =— = ar 6 »"~V' is neither nor infinite : (21) 



s s 



therefore, because u s is finite, by (21), 



e = l ( 5 = 1,2,3,...,*-1) . . . (25) 

 Also, for x K = 0, that is 



^+#2+^3+ . . . +x K _ 1 =l and n K = 0, 



we have, by definition of the derivative, (13), (2), (22), (11), 

 and (12), 



Jjl = N JjL_ = N?? = Pa = -jfrn-lf* is neither nor infinite ;(26) 

 therefore, because u K is finite . . . ., by (21), 



-. = i ( 27 > 



* This is substantially Margules' formula, p Y = Pj x a <> e w , where m = 

 for x = 1 (see Zoc. cit.). There seems to be an idea in the minds of some 

 that this formula involves an assumption ; but, clearly, (21) simply defines 

 the use of the symbol it,. The only assumption that Margules makes in 

 the formula as he gives it is that u can be developed according to positive 

 integral powers of 1— x. 



