Pressures in Liquid Mixtures. 105 



terms of which the #'s, »'s, p's, and the new function co are 

 to be expressed, we have *, from (32), 



Zidfa—uJ 4-%^(«2- w J +^( M 3 — M J+ ■•■ + z K -i d( , u K -i~ u K ) —dm. (35) 



As # s (s = 1, 2, 3, . . ., k — 1) may have any value from 



k — 2 1 

 to 1, z may have any value from -to -\ -, so that 



the absolute value of z s is always less than 1, which is 

 decidedly advantageous when we have to do with infinite 

 series in the variables. Namely, the smaller the absolute 

 values of the variables the more readily is the convergency 

 of such a series determined and the fewer terms will it 

 probably be necessary to use in calculating its value to any 

 given degree of accuracy. Further on we have given a 

 special method for treating a binary mixture, which is 

 practically equivalent to that mentioned in the last footnote ; 

 by means of which we have calculated the formulae (2tf) 

 from actual observations of several binary mixtures. We 

 have also calculated these same formulae in terms of the x's, 

 and the special method not only has the advantage of using 

 variables with smaller absolute values, but gives series for 

 the w's with more rapidly diminishing coefficients than the 

 former. It does not, however, seem possible to predict that 

 the series for the w's will always be more convergent, or have 

 more rapidly diminishing coefficients, the smaller the absolute 

 values of the variables involved. 



* The method we are going to apply to (35) to determine 

 «,, u 2 , u 3 , . . . , u K from co as functions of the z's may be applied to (31) 

 to determine u 1} u 2 , u Sf . . . , u K _f from u K as functions of the x's. Also, 

 if, after changing the signs ol all terms of (31), we had distributed only 



k — 1 K-ths of — du K among the other terms, had added - {du x -\-du 2 -\-du z + 



. . . +du K ) to both members, and had put 



x s = z s (s = 1, 2, 3, ... , k— 1) and - (u x +u 2 +u z + . . . +u K ) = w, 



we should have had the same equation (35) with the new variables z 

 and the new function ©. The determination of u x , u 2 , w 3 , . . ., u K and 

 this function a> in terms of the new s's will follow the same lines as in 

 the text. For a binary mixture (k = 2) this method is preferable to that 

 given above, because here the absolute value of any z never exceeds \, 

 while above it may amount to 1 (for the corresponding x — 0). But for 

 values of k ^3, the method of the text gives the smaller maximum abso- 



k— 2 . k— 1 



lute value of any z, namely r instead of (for the corresponding 



x — 1). 



