( 9'^ ) 



da da 

 d.v, d.t'„ 



da 



d.Vn 



db 



diCn 



A; 



thus out of the system of linear equations 



da db 



o/fj d.i\ 



da db 



— X - — = O , besides a\ ■■]- x.^ -f- . . . -\- .r„ =: 1 



\da; 



de?;. 



da db 



or explicitly 



I («1 — Xb^ ) A'j + (ttj^ — Xb^^) .»2 + + (ain - ;i>i„) .t'„ = O 



( T^) Kl — ^^^2l) '^'l + («2 — >-^2 ) '''ï + + («2» — Xbou) .V'„ = o 



f (a,n — Xbn\) A'i + («.''2 — Xb,,2) .^'2 + + («n — Xbn ) .v„ = O 



and 



•^'i + '^2 + — 4- ''^■« = 1- 



In these equations bpq = ^^^, and a^^j = r/^^, ; they are the same 

 quantities appearing in the theory of the binary mixtures. All a's 

 and ^'s are essentially positive. 



Out of the system ( V) can be deduced that — for the mixtures 



a 

 sought for — — or X can be found out of an equation of degree ?^: 



An =. a.^1 — Aègi a^ — Xb.^ a-2n — Xb^n = 0. 



«Hi — '^^il «h2 — -^^«2 a,i — Xbn 



Algebraically there are thus n solutions — the values of .i- belonging 

 to the obtained values of X are found out of 



X, ■= 



J/„ 



31, 



M,n 



2^ Mu 2^ Mis 22 ^^^'^ 



S=l S=l s=l 



Here AIp^ represents the coefficient of a^g — Xb^g in the develop- 

 ment of A„ according to the elements of the ^>''' row whilst in these 

 expressions we must substitute successively for ;. the roots of A„ = 0. 



If such a mixture is to be realized w^e must find for the mole- 

 cular fractions positive values so that one of the tw^o following 

 systems of inequalities must be satisfied 



