1904-5.] Vanishing Aggregates of Determinant Minors. 853 
Vanishing Aggregates of Determinant Minors. By 
Professor W. H. Metzler. 
(MS. received May 15, 1905. Read June 5, 1905.) 
1. Since a persymmetric determinant is a particular case of an 
axisymmetric determinant, it follows that every type of vanishing 
aggregate for axisymmetric determinants is also a vanishing 
aggregate for persymmetric determinants. The principal object of 
this paper is to give a series of theorems (I., II., III., IV., V.), by 
the application of which to any vanishing aggregate of minors of 
persymmetric determinants new vanishing aggregates are obtained 
(7-12), which, though true for persymmetric, are no longer true for 
axisymmetric determinants. 
It is also shown that certain theorems given by Muir* come 
out as particular cases of a theorem given by the present writer.! 
2. The following types of vanishing aggregates of determinants 
are known : — 
(a) Tor axisymmetric determinants : 
2 
| 1234 
15678 
1234 
1235 
+ 
1236 
1237 
+ 
1238 
1 5678 
4678 
4578 
4568 
4567 
2 
1234 
5678 
§ 
= 0 
(1) 
( 2 ) 
= 0 
( 3 ) 
* Muir, “Vanishing Aggregates of Secondary Minors of a Persymmetric 
Determinant,” Trans. Roy. Soc. Edin., vol. xl., 1902. This will be referred 
to hereafter as Muir, paper I. 
t Metzler, “On Certain Aggregates of Determinant Minors,” Trans. 
Amer. Math. Soc., October 1901. Referred to hereafter as Metzler, paper I. 
X Kronecker, “Die Subdeterminanten Symmetricher Systeme,” Berliner 
Berichte, 1882. 
§ Metzler, l.c., paper I. 
|| Muir, “Aggregates of Minors of an Axisymmetric Determinant,” Phil. 
Mag., April 1902. Referred to hereafter as Muir, paper II. 
