440 Proceedings of Royal Society of Edinburgh. [sess. 
Similarly, taking only the last n - k + 1 equations of the second 
system and eliminating u k x ? Ujc+2 • • • . , u n there is obtained 
E u + K i u i + • • • + \\ = RF^ + RF, +1 ^ +1 + . . . . + RF n t n , 
where the multipliers F fc , F &+1 , . . . , F w by which the elimina- 
tion is effected are 
2±A<*+fA ( *+* ) ...A ( : ) , 
-E±A^X|..A« 
/_ l\ n - Jc 'Z + a k,c+L) a { 
{ 1 / Zj±JrV k A i+l •••‘“n-l* 
and consequently by E, Ej , . . . , E^ are denoted 
2±A,<f. • , A« 
2±a;a ( ^...a<: ) , 
(ft+1) \ (ft +2) (n) 
2±A»a£?...A» 
These two derived equations (X), (Y), however, must be identi- 
cal, because they may be both viewed as giving t k in terms of 
4+i) 4+2 • • ■ , t m u, u v . . . , u k , and, as the first system of equa- 
tions shows, this can only be done in one way. We thus have the 
deduction 
C.-H.F*’ 
ie 2 ±«h'<- . ■ «<*_-» S±A<*>A%?. . .a? 
. .a? bs±a^a»+J...aW- 
This is the keystone of the demonstration. The simple continua- 
tion of it may for sake of historical colour be given in Jacobi’s own 
words (p. 304) : * — 
“ In hac formula generali ipsi k tribuendo valores 
n - 1 , n - 2 ? n- 3 , . . . ,1, 
prodit : 
The demonstration in the original is considerably disfigured by misprints. 
