390 
Proceedings of Royal Society of Edinburgh. 
and taking p v p B , ... , p n as multipliers, lie readily shows of 
course that if the multipliers can be got to satisfy the conditions 
(2)iPi + (2)^2 + (2)sft + .... + ( 2)„p n = 0 1 
(3)iih + ( 3 ) 2^2 + ( 3 ) 3^3 + • • • • 
+ (3 )nVn = 0 
(4)iPi + (4 ) 2 y> 2 + {^)sPz + • • • . 
+ (1)* = 0 
> 
(”)iP: 1 + (») 2 i , 2 + 'Wafts + • • • • 
+ (n)nPn = 0 
i 
the value of x 1 will be 
K 1 P 1 + [^ 2^2 + [l]sPs + • • 
• • + [1 ]nPn . 
(UiPi + (O 2 P 2 + (Usft + • 
• • • + (1 )nVn ’ 
in other words, that x x can be determined at once if a function 
(l)ift + ( 1 ) 2^2 + (OaPg + • 
• • • + (1)a 
can be formed of such a character that it will vanish when instead 
of the coefficients (l) l5 (1) 2 , (1) 3 , . . . 
, (1) M we substitute the 
members of any one of the n— 1 rows 
( 2)1 ( 2)2 (2) 3 . • • 
• (2). 
(3)i (3)2 (3)3 • • • 
• (3)» 
(Cl (C 2 (Cs • ■ • 
• (C» 
Wi (re) 2 (n) s .... ( n)„ ; 
the said function itself being the denominator of the value of x 1 
and the numerator being derivable from the denominator by insert- 
ing [l]i> [U 2 . [1]» [1]» ™ place of (1) 1; (1) 2 , (l) s> . . . ,(!)„, 
Further, as any one of the unknowns may be made the first, 
the complete solution is thus put in prospect. “Alles kommt 
demnach auf die Entwickelung einer Function von der angegebenen 
Beschaffenheit an.” (xiii. 5) 
Two rules, Grunert says, have been given for the construction of 
such a function, one by Cramer, the other by Bezout. The former 
he states, and illustrates by constructing the desired function for 
the case where n = 4. The proof of it is then attempted, and is 
said at the outset to consist essentially in establishing the proposi- 
