1888.] Dr T. Muir on Aggregates of Determinants. 
97 
any axisymmetric determinant are connected by a linear relation, 
or, as I have tried to put it more definitely in the present title, that 
certain aggregates of minors are equal to zero. This discovery has 
attracted considerable attention in Germany, as the list of papers 
herewith given suffices to show.* 
The object of the present communication is, in the first place, to 
point out how much the subject gains in simplicity and clearness, 
if we consider such identities altogether apart from axisymmetric 
determinants ; and, in the second place, to direct attention to a new 
class of identities which have a similar special application, 
2. Let us take then any general determinant of the order, but 
for shortness’ sake let it be written of the 5*^^ order, viz., 
I 1 • 
We have clearly at the outset the vanishing aggregate 
a^ «2 ^3 ^4 ^5 
^2 ^3 ^4 ^5 
(Xi «3 «4 
5 i l>2 ^3 
62 &3 &5 
Cl Cg Cg C^ C5 
- 
C2 C3 C4 Cg 
-b 
^1 ^3 ^4 ^5 
c?i c?2 d^ 
^2 ^3 ^4 ^5 
c?i cfg d^ d^ 
Cl 62 H ^4 ^5 
^2 
a^ a^ a^ 
«1 ^2 a^ 
^1 ^2 ^3 ^4 
&2 ^4 ^5 
&1 &2 ^3 ^5 
62 63 \ 
Cl C2 C^ Cg 
+ 
Cl C2 C3 Cg 
- 
Cl C2 C3 C4 
di d^ d^ d^ 
di d^ d^ d^ 
c?i dc^ c^3 d^ 
^3 
^4 
If we complete the first column of each of the last five determin- 
ants by inserting the elements A, B, C, D, the result is still a 
vanishing aggregate : and if the last row of the second determinant 
be completed by inserting the elements a, y8, y, S, and if, at the 
same time, one of these elements be inserted in the place (5,2) of 
the remaining determinants, no alteration is even then made in the 
value of the aggregate : that is to say, we have the identity 
* Runge, C. “ Die linearen Relationen zwischen den verscliiedenen Subdeter- 
minanten symmetrischer Systeme,” Crelles Joiirn., xciii. pp, 319-327. 
Mebmke, R. “ Bemerkung iiber die Subdeterminanten symmetrischer 
Systeme,” Math. Annalen, xxvi. pp. 209, 210. 
Schendel, L. “ Der Kronecker’sche Subdetermintensatz,” /. Math, 
u. Phys., xxxii. pp. 119, 120. 
VOL. XV. 15/6/88 
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