Pressures in Liquid Mixtures. 99 



then 



*r = p «p = Na >> - 1, 2, 3, . . . *), . (2) 



and 



^ + ^2+^3+ . . . +# K =5 1 (3) 



Furthermore, if dx\, dx. 2 , dx%, . . , dx K are the infinitesimal 

 changes in the x's that correspond to any infinitesimal 

 variation of composition of the mixture, we have, by (3) 



dx 1 + dx 2 +dx z + . . . + dx K = 0. . . . (4) 



On account of (3) the x 9 s are not all independent, as is also 

 implied by (4), but when the values of any tc— 1 of the x's, 

 positive or and having a sum not greater than 1, are given, 

 the value of the remaining x will be determined by (3). In 

 particular, if any x has the value 1, all the other x's have the 

 common value 0, and, if all but one of the x's are 0, that one 

 is 1. If x r =0, we have, by (2), ?i r =0 and vice versa, and the 

 rth component is absent. If x r =l, the rth component is the 

 only one present, and constitutes the whole mixture ; then 

 every n is excepting n r . When we speak hereafter of " all 

 values of the x's," we shall mean all sets of values that 

 satisfy (3). 



By condition c, each of the partial pressures p is a con- 

 tinuous singly- valued function of the x's, finite for all values 

 of the x's. In consequence of (3) it may be expressed as a 

 function of any k — 1 of the x's, and, on replacement of the 

 x's by their values from (2), it may be expressed as a function 

 of the n's. It will be convenient to represent p £ , the partial 

 pressure of the sth component, by p ( s n) when expressed as a 

 function of the n's, by p^ x ' when expressed as a function of 

 all the x's by substitution of the n's in the terms of the ^s 

 from (2) in pf' (N fails out and the result of the substitution 

 is just the same as if each n in p^ were replaced by the cor- 

 responding x, because p^ l) is a homogeneous function of the 

 n's of degree 0), and by p^ when expressed as a function of 

 x x , x 2 , x 3 , . . . x K -y (without x K ). It is to be observed 

 that p^\ p^\ and p^' are perfectly definite expressions ; but 

 p s can be expressed in many ways as a function of all the x's, 

 on account of the relation (3). Then, because p^ l) is a homo- 

 geneous function of the n's of degree 0, 



o>h otto o>h o>i< 



H2 



