The theory of determinants in the historical order of development, by Sir Thomas Muir.

AXISYMMETRIC DETERMINANTS (SEELIGER, 1875) 113

SEELIGER, H. (1875).
[Bemerkungen fiber 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 pth power of an axisymmetric determinant. To this he adds the statement that the
four modes of performing the multiplication lead to the same result.
Another proposition is that the pt" 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 I alb2c3d4 and B3. Since all the two-line minors of
the adjugate must vanish, he has of course
0 AAI = IA1B3 A =- IA 4B
= ICB3 = 1 C2B3- IC43
=DB ID S (DB31 = IDB3a D
and.'. 0 == AB1 = A3 B = A3B
- 03B1 = 03)32 0 (3B4 r
= D3B1 = D3B2 = D3B,
from which it is easy to see that
either 0 = A3= 3 = D3,
or 0 = B1 = B= B4.
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
row of the adjugate with M must also vanish, or those which are in the
same column. As a corollary it is added that when in addition A is
axisymmetric and M is coaxial, there is no alternative.
Following on this is a proposition less readily acceptable, namely:
If an n-line axisymmetric determinant and n-2 of its primary coaxial
minors simultaneously vanish, then all the other primary minors
vanish also. It is at once seen that after the application of the
preceding corollary the only elements of the adjugate that require
M.D. 11. I