1899 - 1900 .] Dr Muir on Determinant Minors. 
145 
1 2 3 ! 
1 2 3 
123 
568 
578 
678 
1 2 3 j 
123 
1 2 3 
4 6 7) 
468 
47 8 
123! 1 1 2 3 | 
45 7! I 4 5 8 j 
123 
I 4 5 6 
used twice over, the accompanying sign being changed in the 
case of the second occurrence. The resulting expansion thus is 
1 2 3 /4 8\ 
5 6 7 \8 _ 4/ 
1 2 3 I /4 _ 7\ 
5 6 8 I \7 V + 
123/4 
57 8 \6 
123 /4 5\ 
6 7 8 \5 *"* 4/ 
123/5 
467 \8 
123/5 
46 8 \7 
12 3 /5 _ 6\ 
47 8 V6 5) 
123 
457 
6 _ 8 
8 6 
123 
458 
1 2 3 /7 _ 8\ 
45 6 \8 7/ 
(4) The general theorem to which we are led is, of course, 
readily enunciated when the law of formation of the terms on 
the right-hand side of the identity is grasped. This is most 
easily done by first considering the two triangular arrangements 
of elements which go to form the second factors of the terms. 
The parent determinant being of the (2 n) th order, the first of 
these arrangements is 
n n 
n 
n 
2 n 2n-\ 
2n-2 * 
n + 1 
n+ 1 
n + 1 
w + 1 
2 n 
2n - 1 * 
n 2 
n + 2 
n + 2 
2 n " 
n + 3 
2n - 1 
2 n , 
VOL. XXIII. K 
