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

SKEW DETERlMINANTS (CAYLEY, 1847)

261

Xaj3X22 - X2XcXa2,
XP2 Xa2
=[32].[a2];
consequently " Le de'terrninant gauche et syrnvtrique qu'on obtient en donnanrt d r les valemrs a,2,3,...,n, et d s les vleumrs
P,2,3,... n (ov'> n est pair) se re'duit a'
[a 2 3... n] - [O 2 3... n]
et en pai-ticmuli r, en domrnctnt d r, s les vcalers 1,2,..., n ce
ditermninant se re'dit a' [1 2 3... n]2."
Going back now to the expansion of the skew determinant Q2
with which the paper opened, and taking for simplicity's sake*
X,.. = 1 in every case, Cayley readily obtains,
for n even, Q = [123....]2
+ [34...n]2 + [24... n]2 +
+ [56...n]2+...
+ 1,
and, for n odd, Q = [23... n]2 + [13....n]2 +...
+ [4... n]2
+.
A special example of each identity is given, namely, the
examples in which vn=4 and 3 respectively. If we make a
slight change in the left member, viz., write Q2 in Cayley's
vertical-line notation (which, by the way, considering the help
it would have given, and the fact that it had been introduced
six years previously, it is surprising not to find employed in
this paper), these examples take the form,1 X12 X13 X 14 1 12 13 14
X121 23 X24 or -12 1 23 24
\ L13 X23 1 X34 -13-23 1 34
X14 -X24 -X04 1 -14- 24- 34 1
(X12X34 - X13X24 + X14X23)2 + N212 + X213 ~ N214 + X234 +?224 + X223 + 1 >
= [1 2342 + [12]2 + [1312+ [142+[34]2 [2412 +[2312 + I;
* And of course without loss of generality, as Cayley might have said.