Pressures in Liquid Mixtures. 109 



written in the left member of (44) is the coefficient of 



-$l JT* ~/7 3 ^r" 1 ~?*-l 



"1 ^2 "3 ••' *r '*• ~/c-l 



in the A>th term of the left number of the r-th equation (43) 

 and the right member of (44) is the coefficient of the same 

 product of powers of the z's in the right member of the 

 r-th equation (43). But it is to be observed that the 

 numerical multiplier of the r-th term of the r-th equation 

 (44) (the term for which k = r) is g r — 1 and not g r , and that 

 the &-th term of the left member of the r-th equation (44) 

 is to be omitted if the &-th g of the set in question (that 

 is g k ) is 0. The r-th equation (44) falls out if g r = for 

 the set in question. 



The left members of equations (43) have no constant 

 terms and, therefore, the derivatives of co with respect to 

 z i> z 2i Z & • • • j z -l nave no constant terms ; that is 



c m = (4 = 1,2, 3,..., «-l). . . (45) 



where c 1(k) denotes the coefficient of z k in o> (the c whose 

 suffixes are all except the /c-th and that is 1). Further- 

 more, if a { p and c are used for brevity to denote the 

 constant terms of u r and o>, respectively, we have, by (34), 



4 ) + af> + af> + ...af- 1) =( l c-l)c . . . (46) 



If we write, for any given set of values of g u q 2s g d . . . . , ff K _ l9 



^1+^2+^3 + ... + ^_ r =G . , . (47) 



and add the system of r equations (44) for this set, remem- 

 bering what we said about the numerical coefficient of the 

 term of each equation that corresponds to the number of that 

 equation in the system, we find 



On subtracting G — 1 times the r-th equation (44) from 

 g r times equation (48) we have 



(s-i)(«!Vi..-«V...)=^ 



or, writing g r +l instead of g ri preserving the notation (47), 

 G («<?... -«!?...) = (</,. + !) «,*,+!.. ('' = 1, 2, 3,. . ,«-l) . (49) 



