1899-1900.] Dr Muir on Determinant Minors. 
151 
of the parent determinant, 
say, and expand each 
From this, as before, we remove the restriction as to the form 
123456 
123456 
minor of the aggregate in terms of three elements and their co- 
factors, the three elements in the six cases being those of the 3rd 
row 3rd column, 2nd row 2nd column, 1st row 1st column 
respectively. The result of this is 
14 5 1 _ 456 426 _ 456 3 5 6; _ 456! 
1456 “ 451 + 4 5 6 " 426 + 4 5 6!” 356 
45 
1 
45 
1 
45 
| 1 
45 
6 
46 
5 
56 
“ 
45 
*6 “ 
46 
* 5 + 
5 6 
;*4 
45 
*1 + 
45 
*1 
45 
46 
2 
46 
| 2 
46 
I 2 
45 
6 
46 
5 
56! 
45 
* 6 + 
46 
‘5 “ 
5 6 
* 4 + 
46 
’2 
46 
*2 + 
46j 
5 6 
3 
56 
3 
56 
3 
45 
6 
4 6 
5 
56 
+ I 
45 
'6 
46 
*5 + 
5 6 
*4 
5 6 
-3 + 
5 6 
*3 ” 
56 
4 
1 
4 
2 
4 
3, 
where, it is worthy of notice, each of the three lines on the right- 
hand side of the identity contains the expansion of two minors 
which are conjugate to one another, this arrangement being made 
for the purpose of showing more clearly that the eighteen two- 
lined minors which appear in the expansion, consist merely of the 
45 6 
nine such minors formable from 
4 5 6 
nine occurs first with one of the elements of 
and then with one of the elements of 
, and that each of these 
as a cofactor. 
This suffices to 
1 23 
456 
456 
1 2 3 
draw attention in passing to the fact, which can also be reached 
by consideration of the left-hand side, that the identity involves 
all the elements of 
and that each element of 
123456 
123456 
456 
except those of the minor 
123 
123 
element of 
123 
456 
,456 
and its conjugate 
occurs four times, while each 
4 56 
1 23 
occurs only once. 
It is thus seen that the right-hand side may be condensed into 
