1888 .] Dr T. Muir on Aggregates of Determinants. 
99 
1,4 2,4 3,4 
5.6 5,7 5,8 
6.7 6,8 
7.8 
so that the identity is 
15 
16 
17 
18 
14 
16 
17 
18 
14 
15 
17 
18 
25 
26 
27 
28 
24 
26 
27 
28 
24 
25 
27 
28 
35 
36 
37 
38 
34 
36 
37 
38 
+ 
34 
35 
37 
38 
45 
46 
47 
48 
45 
56 
57 
58 
46 
56 
67 
68 
14 
15 
16 
18 
14 
15 
16 
17 
24 
25 
26 
28 
24 
25 
26 
27 
34 
35 
36 
38 
+ 
34 
35 
36 
37 
47 
57 
67 
78 
48 
58 
68 
78 
A glance at this suffices to show that the second determinant here 
would be a minor of any determinant of the 8^ order, in which the 
elements 54 and 45 were identical: that the third determinaot 
would likewise be a minor of the same determinant, if 
46 = 64 and 56 = 65; 
also the fourth determinant, if 
47 = 74, 57 = 75, 67 = 76; 
and the fifth determinant, if 
48 = 84, 58 = 85, 68 = 86, 78 = 87 . 
Now in the axisymmetric determinant 
11, 22, 33,...., 88 
all these conditions hold. Consequently the above identity is an 
identity connecting five of the minors of 
I 11, 22, 33,...., 88 
and this is Kronecker’s theorem. 
4. It has been said that the number of elements occurring in the 
identity is ^(n+ 1) (on - 2), ?i being the order of the determinants 
involved. When, therefore, the identity is given in connection with 
an axisymmetric determinant of the ^n^^ order, which, as we know, has 
1) distinct elements, it is suggested to inquire which elements 
of the latter do not occur. Their number evidently is 
n(f.n + 1) - J(^^ + 1 ) if>n - 2) 
i.e. + -^n + 1 
