1896 - 97 .] Dr Muir on Bordered Skew Determinant. 
355 
orders, becoming equal. For example, when the order is the 5th 
the product 
a 2 a 3 
Pi P 2 
7i 
m 1 7q h 2 7< 3 h A 
h 1 h 2 ti 3 h 4 
a l a 2 a 3 
/^1 P'2 
* , 
on account of the identity of the first two frame-lines of the second 
Pfaffian, becomes 
m i I a i a 2 a 2 2 
Pi P-2 
7i ; 
and the full statement of the identity is 
m n 
7q 
h 3 
h. 
K 
^2 
K 
h 
= m l | oq a 2 a 3 
m 2 
a l 
a 2 
a 3 
Pi P2 
~ a i 
m 3 
Pi 
P2 
7i 
~ a 2 
-Pi 
m 4 
7i 
+ 2 
— a 0 
\j 
-P2 
” 7i 
m 5 
or = ! cq a 3 
Pi P 2 
7i 
+ I ^2 ^3 \ 
Pi P 2 
7l 
2 
•yi 2 + 2 w 2 m 3 m 4'^4 2 + ni l m 2 m 3 m 4 in 5 , 
2 + 2 ? ^i^2 ?w 3'7i 2 
+ m 1 m 2 m 3 m^m 5 , 
if we bear in mind the wider sphere to which 2 now refers. 
This special case is, as implied in § 1, that first dealt with by 
Cayley, being the subject of his second paper on Skew Determin- 
ants.* 
14. A consideration of this expansion of a skew determinant 
suffices to determine the number of unlike terms in such a deter- 
minant. For example, for the 5th order we clearly have the 
number of unlike terms 
= ~ ^3>i) + C 5 , 3 4 - C 5 , 5 , 
= 30 + 10 + 1 , 
= 41 . 
* Cayley, A., “ Sur les determinants gaudies,” Cr die’s Journ., xxxviii. 
pp. 93-96 ; or Collected Math. Papers , i. pp. 410-413. 
