59 
1912-13.] Dr Muir on Axisymmetric Determinants. 
and Cayley establishes it by taking his own, namely, 
u n -nu n _ x + \(n- l)(w- 2)%_ 3 = 0 
or, say, E n = 0 , 
and showing that the other is 
E n + (w-l)E n _ 1 = 0. 
The first eight values of u n he finds to be 
1,2,5,17,73,388,2461,18156, .... 
Roberts himself identifies the problem with that of finding “ the 
number of distinct ways in which 2 n things, two of a sort, can be made 
into parcels of 2.” 
Seeliger, H. (1875). 
[Bemerkungen liber symmetrische Determinanten, und Anwendung 
auf eine Aufgabe der analytischen Geometrie. Zeitschrift f. 
Math. u. Phys., xx. pp. 467-474.] 
With the help of an unwieldy multiple-sigma representation of the 
elements of a power-determinant Seeliger arrives at Sylvester’s unproved 
proposition of 1852 regarding the p th power of an axisymmetric deter- 
minant. To this he adds the statement that the four modes of performing 
the multiplication lead to the same result. Another proposition is that 
the p a power of a vanishing two-line determinant is axisymmetric. 
He next investigates the consequences of the simultaneous vanishing of 
a determinant and one of its primary minors, say the determinant | af^c^d^ \ 
and B 3 . Since all the two-line minors of the adjugate must vanish, he has 
of course 
0 = j A 1 B 3 1 = A 2 B 3 1 = | A 4 B 3 1 ) 
= | CjB 3 | = | C 2 B 3 1 - | C 4 B 8 1 V 
-I DA I = I D 2 B 3 1 = |D 4 B 3 I) 
and 0 = A 3 B x = A 3 B 2 = A 3 B 4 ) 
= c 3 b 1 = c 8 b 2 = c 3 bA 
“ ^3 ^1 = ^3^2 : B 3 -^4 ) 
from which it is easy to see that 
either 0 = A 3 = C 3 = D 3 , 
or 0 = B x = B 2 = B 4 . 
The general result is that if a determinant and one of its primary 
minors, M, vanish, then the other primary minors which are in the same 
