( Ö9 ) 



A\ L\.. . L\,^2 Ln i Ln F ^ A, L\ L\... A% _2 A,._i A„ L\ + 

 + A,A,A% . . . A%_2A„_,A„L%4-A%A,A,A% . . . A^_oA„_,A„L% + 

 + . . . -f A% A% . . . AV -2 A^^i A/, AV+i . . . A^,_o A„_, A„ L\^ . . . 



. . . A% A% . . . A%_2 /A ^). 



Here is 



-AA] ffi,, — ).h 



L. 



p—u 



l,=-z 







a,^ — ).b, 



ao l ^'^2 1 (t2/; XOq/i 



• (M,.s — i — •^''m.s— 1 



• «2, s— 1 — ^^'2.s — 1 



'P 

 n\p 



7' 





•1, 1 



So we find 



■;ai 



«s— l,,s-l — Xbs~\,s—\ as—\,j, — ?.bs—\,p 

 «s, s— 1 — Xbs,s^l O.S,, — Xb^j, 



A, "^A,A. 



H- 



Zjijj .6';,. 

 ^p—\^p 



+ 



iv'« 



t-^n—l^n 



% 



For a stationary point A„ becomes equal to 0, and so tiie last 



term disappears as it is 



A^ 



.^. ^). 



1) For the deduction a continual use has been made of the theorem : 



A AeeeA A - A A 



p (/ r s ji s (j r 



PV 

 r s 



Weber: Lehrbucii der Algebra I, 2"'^ edition, p. 115. 



2) In connection with the following it is easy to find back out of this the system (V). 

 Gomp. for a ternary system: van uer Waals, Proceedings Royal Acad, of Sc. 

 190^2, Vol. 5. p. iJ35. 



'*) Without looking more closely at the breaking up into squares, as is performed 



above, l^iouxn shows in his Rigid Dynamics, Advanced Part, 5th ed., p. 426 



Ay) 

 with the help of invariants that the coetticient ot Lp has the sign of ., which 



Ay,_i 



agrees to the above. 



*) There is a ditliculty however when for A = Aq also A«— i becomes 0, so when 

 a stationary point in the mixture (l,2,...w — 1) becomes at the same time a statio- 

 nary point for the total mixture. When breaking up into squares we then shall 

 change the order of succession in order to avoid x,r presenting itself as last term. 

 If all first minors belonging to elements out of the principal diagonal A" are Ü 

 then all minors must be 0, if there are to be stationary points. The equations 

 (V) have then a higher degree of dependence and there are an infinite number of 



mixtures for which - becomes stationary, 

 o 



7* 



