1903 — 4 .] 
Dr Muir on General Determinants. 
77 
and the system of equations, obtained by equating each of 
these determinants to zero, by the notation 
( 3 ) 
X Y X 2 .... x n 
Aj A2 .... A^ 
= 0 .” 
Ki K 2 
I 
A theorem is next enunciated in regard to the expression of 
any one of the determinants in terms of n - q of them. 
“ The ~~ ^ ^ 
1 • 2 • • • (n - q - 
formula reduce themselves to n-q independent equations. 
Imagine these expressed by 
(1) = 0 , (2) = 0 , . . . . , (n — q) — § , 
1) 
equations represented by this 
any 
one of the determinants is reducible to the form 
©i(l) 4- ® 2 (2) + • • • + ®n- q { n -9) 
where © x , © 2 , . . . , © n _ ? are coefficients independent 
of , x 2 , . . . , x n ” 
No proof is given. 
The introduction of the notation is fully justified by two 
theorems which follow. The first is virtually to the effect that 
we may multiply both sides of (3) by the determinant 
*2 
* * * 
K 
ft 
/* 2 
* 
T 1 
t 2 
. . . 
T n 
just as if (3) were a single equation instead of C n , q +\ equations, 
and as if the left-hand side were a determinant ; and the result, 
written in the form 
4- 
rH 
< 
* + K- r n 
fh x i "** ' ‘ 
* “t" [A n X n ‘ * * 
t i x i + ■ ■ 
1 ' d" T n X n | 
(6) 
XA x + • • 
• + ^-n A w 
/xjAi 4 
• + [l n A n • • • 
T 1 A 1 + • • 
• + r n A n \ 
+ ‘ 
• • 4 - A,K 1( 
^i K i + * 
* 4" * • 
T 1 K 1 + • • 
. . . 
h Tn^-n 
