1888 .] Dr T. Muir on Aggregates of Determinants. 
101 
and thereby preventing them occurring among the numbers of the 
columns. 
Further, we readily conclude that as there are fifteen elements 
outside the main diagonal of D, there must be connected with D 
fifteen identities like the above, the four determinants of each 
identity being easily got as soon as the element to be omitted has 
been decided upon. The number of different determinants involved 
in the fifteen identities is the number of ways in which the numbers 
1, 2, 3, 4, 5, 6, can he separated into two sets, and therefore, is 
^Cg ,3 ^.e. 10. Of these fifteen identities connecting ten determin- 
ants, it will be found that only five are independent. 
6. Further, ''each identity consists of 4 x 6, i.e. 24, terras separable 
into 12 pairs, the terms of each pair being equal in magnitude, and 
opposite in sign. Now it is a curious fact, that the twelve different 
terms are exactly the twelve terms of the Pfaffian got from D, by 
deleting the elements which are not found in the identity. For 
example, the twelve different terms of the typical identity given 
above, are exactly the twelve terms of the Pfaffian, 
I 0 a^ 
&3 b^ 
C4 Cg Cj. 
^6 i 
so that, in fact, the identity may be put in the form 
(X 3 a^ «g 
^3 ^4 ^6 
«3 «4 a 
5 
&4 ^5 b^ 
- 
h h h 
+ 
h ^4 h 
- 
h ^4 
5 
C 4 Cg 
dg 
^5 ^5 ^6 
e, 
6 
. «3 
a, 
1 ^5 ^6 
• 1 . % 
«4 
h 
h 
: h h 
1 \ . 
h 
h h 
^4 
^4 
^5 ^6 
d^ d^ 
cZg f?g 
% 
1 5 
Apparently this is equally true for all the higher orders. Thus, in 
the case of the axisymmetric determinant, 
111, 22,33,....,88 L,=, 
