154 
Proceedings of Royal Society of Edinburgh. [sess* 
y 
n+1, 
n + 
2 , 
/u,=l, . . n 
n+1, 
n + 
2 , 
2 
n + 1, 
n + 
2 , 
ju=l, . . n 
n+1, 
n + 
2 , 
V 
n + 1, 
n + 
2 , 
a 
r 
+ 
11 
n+1, 
n + 
9 
"j 
n -/JL+ 1, . . 2n 
n + /x , . . 2n 
n + fx , . . 2n 
n-//,+ l, . . 2n 
n— /* + 1, . . 2n 
n + /x , . . 2n • 
(11) From this there follows, exactly as before, a variant form 
of the enunciation of the less general theorem with which we 
started, viz., 
In connection with an even-ordered determinant 
1 2 3 . . . 2n 
1 2 3 . . . 2n 
which is such that the elements of 
order identical with those of 
1 2 
n+1, n + 2, 
2n, . . ., n + 2, n + 1 
n 2 1 
n j 
. ., 2n | 
we have 
are m 
2 
n+ 1, n + 2, 
n + 1, n + 2, 
n - 1, . . ., 2n 
n + /JL , . . 2n 
- 2 
n + 1, n + 2, . . ., n + ^ , . . 2n j Q 
n + 1, n + 2, . . ., n-/x+ 1, . . ., 2n | 
The advantage of this form of enunciation, again, is that it 
indicates the limited amount of centro-symmetry which is 
necessary for the validity of one of my 1888 identities, and shows 
that the number of such identities possible, when the centro- 
symmetry is complete and the parent determinant is of the (2w) th 
order, is C 2 n,n- 
