718 
Proceedings of Royal Society of Edinburgh. [sess. 
combination ^ + S 1 6 ^ are taken from the given 2 r numbers, etc. 
V a i <V 
In what follows we will use the umbral notation and be con- 
cerned with minors of the form 
I (*i)(« 2 ) • • • • I 
' (2/1X2/2) • • • • 
where (^ 1 )(^ 2 ) • • ■ • (^ 1 X^ 2 ) • • •• . is a selection of the 2 r given 
numbers. 
The sign factor of any term in an aggregate is positive or 
negative according as the number of inversions in (x 1 )(x 2 ) .... 
(y- l )(y 2 ) .... is even or odd. 
Let s + 1== s' 4- 1' = r, then in the aggregate of minors of order r 
of any determinant A 
2 
\(2r\ 
t + A/ 1 1 + s 1 7*\ 
V 
a i / \ a i i ) 
(M 
V 
a, i) 
certain terms will have the column numbers 
2r\f + s \s\ the 
“1 “2/ 
same 
for all, and other terms will have the column numbers (^ r ^ + s I s \ 
V «i a 2 / 
the same for all, so that we may write 
vliere 
2r 1 1' + sV2r \t' + s I r 
a i A 
2r 1 1' + s | r 
a 7 i 
A =2* 
a n l 
= A + B + C . 
(■2r 
| t f + 
)( 2r 
1 1 + s | s | 
A 
a l / 
K 
a j a 9 
V 
(2r 
1 1 + * 
i«Y 
'•2r\t’ + s 
1 * Sk 
V 
a l 
a 2/V 
a i 
a 2*7 
/2T 
1 t' + S N 
)f 2r 
\_ y ±_ 
1 02 
+ 
> Vi 
A 
a l ' 
K 
a l a 2 
V 
C 2r 
i t' + s 
Is V 
'2r 1 1' + s 
l*i A 
\ 
a \ 
a 2 /\ 
^ a i 
°2 V 
