349 
1896 - 97 .] Dr Muir on Bordered Skew Determinant. 
When we come to partition the groups of (E 2 ), we find it im- 
practicable in the case of one group ; hence the break in the series 
which gives the number of terms in the various developments, 
viz., 
2\ 2 2 , 23, 2S 2 5 - C 5 , 4 . 
7. Of these different developments it is the second (E 2 ) which 
is useful in connection with the proposed Pfaffian development of 
a bordered skew determinant. 
Let the said determinant be 
m 1 
h, 
A 
A 
\ 
-K 
m 
2 a l 
a 2 
a 3 
- k 2 
Wig 
A 
$2 
- k 3 
~ a 2 
A 
m 4 
7i 
-K 
~ a 3 
“ @2 
~7i 
m 5 
and let 
us at once 
apply 
(A) 
. The result is 
h 2 
h 3 
K 
+ 5>2 
m 4 
A 
5^ 
to 
-K 
. 
a 
1 
a 2 
a 3 
— k 2 
© 
A $2 
— k 2 
~ a \ 
• 
A 
A? 
— k 3 - 
-A 
• 7i 
~ A 
~ a 2 
-A 
• 
7i 
-K - 
■ A 
"7i • 
— k± 
~ a 3 
-fi 
2 
“7i 
• 
+ 2>2 m 3 
m 4 
A 
A 
+ 2 m 2 m 3 m 4 
— A 
e 
7i 
e 
r-^ 
I 
- A 
~7i 
® 
+ m 1 m 2 m 3 m i m 5 . 
But every determinant here is a bordered zero-axial skew deter- 
minant, and therefore, by Cayley’s special case, is expressible as 
the product of two Pfaffians, the expansion thus becoming (§ 2) 
a 4 a 2 a 3 
• | m 4 7ij /i 2 A 3 7^ 4 
+ 2^2 1 A A A 
• | k 2 k 3 /r 4 1 
A A 
A A A A 
A A 
A A 
7l 
a l <* 2 a 3 
7i 
7i 
A A 
7i 
+ y 1 1 m l h 3 7 i 4 
k 3 A' 4 
+ 2ra 2 ra 3 m 4 • + m l m 2 m 3 m i m b . 
7i 
