1904-5.] Prof. Metzler on Axisymmetric Determinants. 717 
Variant Forms of Vanishing Aggregates of Minors of 
Axisymmetric Determinants. By Professor W. H. 
Metzler. 
(MS. received March. 13, 1905. Read May 1, 1905.) 
1. Vanishing aggregates of determinant minors have of late 
received considerable attention and are now becoming well known. 
Prom aggregates for axisymmetric determinants of the form given 
by Kronecker,* * * § Muir,f or the present writer J we may get other 
forms, and it is the object of this paper to consider some of these 
new forms, and in particular to show that the general theorem 
given by Professor Nanson § is readily obtained from the vanishing 
aggregates of the type given by the present writer. 
( *2iV 1 1 f - 4 - 
2. Let us here as elsewhere || use ^ 1 ^ j to denote the c^th 
combination (t f + s) at a time of 2 r given numbers, (^ r 1 ^ + S I s \ to 
' a l a 2/ 
denote the a 9 th combination s at a time of the numbers in the 
combination 
the combination 
r 1 t' + s\ /2r 1 1' + s 
2r 1 1' + s 
to denote the complementary of 
, that is, the combination of the re- 
maining numbers after the numbers in the combination 
are taken from the given 2 r numbers, and 
2 r It' + 8 
to denote 
/2r 1 1' + s | s 
V «i <* 2 
the combination of the numbers left after the numbers in the 
* “ Die Subdeterminanten Symmetricher System,” Berliner Berichte , 1882. 
f “Aggregates of Minors of an Axisymmetric Determinant,” Phil. Mag., 
April 1902. 
X “ On certain Aggregates of Determinant Minors,” Trans. Amer. Math. 
Soc., October 1901. 
§ “Minors of Axisymmetric Determinants,” Amer. Jour, of Math., 
January 1905. 
|| Loc. cit ., § 2. Amer. Jour, of Math., vol. xxii., No. 1. 
