142 Proceedings of Royal Society of Edinburgh. [sess. 
On certain Aggregates of Determinant Minors. 
By Thomas Muir, LL.D. 
(Read March 5, 1900.) 
(1) Two curious identities have been established regarding 
certain aggregates of minors of special determinants ; the first, 
which concerns axisymmctric determinants, having been discovered 
by Kronecker in 1882,* and the second, which concerns centro - 
symmetric determinants, having been published by me in 1888.f 
When we come to think of the possibility of generalising these 
identities, it is readily seen that there are at least two lines of 
attack which suggest themselves on reading the mere description 
of the kind of identity; for, in saying that the identities 
deal with “an aggregate of minor determinants of a special de- 
terminant, 5 ’ we are conscious of two points of limitation in the 
description, the one signalised by the word “minor” and the other 
by the word “special.” If, therefore, an identity were obtained 
regarding an aggregate of which the terms were determinants 
unrestricted by a family relationship, we might have one form of 
generalisation ; and if, while retaining the family relationship, we 
succeeded in removing the restriction as to the form of the parent, 
a generalisation of a different type might be the result. 
The former of these lines of attack I have followed up on a 
previous occasion ; in the present paper I take the latter line. 
(2) Kronecker’s theorem, it will be remembered, is to the effect 
that the aggregates 
12 
341 
123 
1 + 661 
11 2 3 41 
| 5 6 7 8 1 
1 1 3 I 114 
| 2 4 j j23, 
1 2 4 
+ 
1 2 5 
3 5 6 
346 
112 3 5 
1236 
4 6 7 8 
+ 
4 5 7 8 
112 6 
1345 , 
1 2 3 7] 1 2 3 81 
| 4 5 6 8 | + 4 5 6 7 |, 
* Kronecker, L., “Die Snbdeterminanten symmetrischer Systeme,” 
Sitzungsb. d. 7c. ATcad. d. Wiss., 1882, pp. 821-824. 
t Muir, T., “On Vanishing Aggregates of Determinants,” Proc. Roy . 
Soc. Edin ., xv. pp. 96-105. 
