1888.] Dr T. Muir on Aggregates of Determinants, 105 
10. Keturning now to § 7, and taking the originating determinants 
A and B in the form 
41 
42 
43 
46 
45 
44 
51 
52 
53 
56 
55 
54 
61 
62 
63 
) 
66 
65 
64 
that is to say, with their elements in the umbral notation_, we 
obtain the identity in the form 
41 
45 
44 
46 
42 
44 
46 
45 
43 
51 
55 
54 
+ 
56 
52 
54 
+ 
56 
55 
53 
61 
65 
64 
66 
62 
64 
66 
65 
63 
41 
42 
43 
46 
45 
44 
46 
45 
44 
56 
55 
54 
+ 
51 
52 
53 
+ 
56 
55 
54 
66 
65 
64 
66 
65 
64 
61 
62 
63 
hTow the first three determinants here are minors of any deter- 
minant of the 6*^ order, and the second three would he minors of a 
determinant of the 6*^ order if its elements were such that 
41, 42, 43, 51, 52, 53, 61, 62, 63 
= 36, 35, 34, 26, 25, 24, 16, 15, 14, 
— in other words, were such that r,s=7-r,7-s. But this is 
exactly the definition of a centro-symmetric determinant. Conse- 
quently the above identity is an identity connecting six of the 
minors of the centro-symmetric determinant 
1 11, 22, 33, . . . , 66 Irs-7-r,7-«) 
and thus we have in regard to such determinants a theorem quite 
co-ordinate with Kronecker’s regarding axisymmetric determinants. 
In the notation of § 5 it would stand as follows: — 
^ I 11, 22, . . . , 66 I he centro-symmetric, i.e. if its elements he 
such that in every case r,s = 7 - r,7 -s, then 
4 
5 
6 
+ 
4 
5 
6 
6 
+ 
4 
5 
1 
4 
2 
4 
5 
1 
4 
2 
6 
4 
5 
6 
+ 
4 
5 
6 
+ 
6 
6 
6 I 
6 I 
