100 
Proceedings of Pioyal Society of Edinburgh. [jan. 6, 
Now among these the 2n elements of the axis of symmetry (or 
main diagonal) of the determinant must be included, as from the 
law of formation of the determinants of the identity their presence 
in the determinants is impossible. Consequently the number of 
elements outside the main diagonal of an axisymmetric determinant 
of the order, which are not involved in Kronecker’s identity, is 
- | 9 ^ + 1 , 
Le. \(gi - 2) (n-V) , 
i.e. 1 + 2 + 3 + . . . . + (gi — 2) . 
For example, when 2?^ = 6, the number not involved is 1 only: 
thus, in the case of the axisymmetric determinant of the 6**" order 
the typical identity is 
Oj ^ ^5 ^6 
d^ d^ d^ 
d ^ 65 
or D, say. 
^4 0^5 Ug 
Ug <2g 
^6 
Ug 
^4 ^5 ^6 
- 
^3 ^5 ^6 
hg b^ 5 g 
- 
h ^4 h 
C4 Cg Cg 
C4 d^ dg 
Cg d^ 6 q 
% ^6 ^6 
which involves all the elements outside the main diagonal, except a^. 
5. Denoting by 
1 2 3 
4 5 6 
that minor of D whose elements belong to the 1®*, 2”*^ and 3’’*^ rows, 
and 4^^, 5*^ and columns of D, we may write this identity in the 
more convenient form 
1 2 3 
4 5 6 
1 2 4 
3 5 6 
1 2 5 
3 4 6 
1 2 
3 4 
It is then easily seen that the exclusion of the main diagonal 
elements is accomplished by making the numbers of the rows 
(e.g. 1,2,5), different from the numbers of the columns (e.g. 3,4,6) : 
and that the exclusion of the one other element U 2 , which occurs in 
both the 1®‘ and 2'^'^ columns of D, is accomplished by always 
including the numbers 1 and 2 among the numbers of the rows. 
