198 Proceedings of Royal Society of Edinburgh. [sess. 
by the translation of the first column to the last place. The 
determinant in this typical term is then further transformed into 
the square root of the product of two determinants like that in 
the term preceding it, the steps of the reasoning being — 
32 
34 
3 n 
2 
23 
24 . . 
. 2 n 
32 
34 . 
. . 3 n 
42 
* 
4 n 
_ 
43 
* 
4 n 
42 
* 
. . 4 n 
n2 
n4 ... 
* 
?z3 
n4 
* 
n2 
ni . . 
* 
= 
* 
24 . . , 
. 2 n 
* 
34 . . 
. 3 n 
43 
4 n 
42 
* 
4 n 
nZ 
n4 . . . 
* 
?z2 
n4 . . 
* 
• j 
the deletion of 23 and 32 in the last step being warranted by the 
fact that their cofactors are determinants similar to the original 
but of odd order n - 3, and therefore have the value zero. The 
development as thus changed has the form of the square of a 
polynomial ; and consequently by extracting the square root there 
results 
1 * 
12 .. 
. . In 
i 
* 
34 . 
. . 3 n 
1 
* 
45 .. 
. 42 
21 
* 
. . 2 n 
= 12- 
43 
* 
. . 4 n 
+ 13- 
54 
* 
. 52 
nl 
n2 . . 
* 
nZ 
n4 . 
. . * 
24 
25 . . 
* 
This, according to the point of view, will be recognised either 
as Cayley’s theorem that an even-ordered skew determinant with 
zeros in the principal diagonal is a square , or as the theorem in 
Pfafhans formulated by Cayley and which in Jacobi’s notation 
would be written 
[123 . .. ro]=12 [34 . . . w] + 13 [45 . . . tz 2] + 14[56 . . . w23] + . . . 
The rest of the section or chapter deals with Cayley’s exten- 
sion of this to skew determinants whose principal elements are 
not zeros, the notation employed being the same. 
