194 Proceedings of Royal Society of Edinburgh. [sess. 
and, for n odd, 0= [23 . . . ?z] 2 + [13 . . . nf + . . . . 
+ [45 . . . nf + 
+ 
+ 1 . 
A special example of each identity is given, viz., the examples in 
which n = 4 and 3 respectively. If we make a slight change in 
the left member, viz., write O 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 
*12 
*13 
*14 
1 
12 
13 
14 
~ *12 
1 
*23 
*24 
or 
-12 
1 
23 
24 
— *13 
“ *23 
1 
*34 
-13 
-23 
1 
34 
“*14 
*24 ~ 
T* 
CO 
1 
1 
-14 
-24 
-34 
1 
- (* 12*34 * 13*24 + *] 4 * 23) 2 
+ ^ 2 i2 +■ ^ 2 13 + * 2 ].4 + ^34 + ^24 + * 2 23 4" 1 j 
= [1234] 2 + [12] 2 4- [13] 2 + [14] 2 + [34] 2 + [24] 2 + [23] 2 + 1 ; 
and 
1 
*12 
*13 
1 
12 
13 
*12 
1 
*23 
or 
-12 
1 
23 
-* 1 S 
1 
CO 
1 
-13- 
23 
1 
~ ^23 4" AAg + A. 2 i2 + 1 . 
= [2 3] 2 + [13] 2 + [12] 2 + 1. 
SPOTTISWOODE (1851, 1853). 
[Elementary Theorems relating to Determinants. By 
William Spottiswoode, M.A., of Balliol College, Oxford, 
viii+ 63 pp. London, 1851. Second edition, as an article 
in Grellds Journ ., li. pp. 209-271, 328-381. ] 
In this the earliest of modern text-books on Determinants, a 
special section (§ ix. pp. 46-51 ; or § vi. pp. 260-266 in second 
edition) is set apart with the heading “ On Skew Determinants.” 
As a matter of fact, however, it is only the latter half of the 
section which at present concerns us, as the other half deals in 
