358 
Proceedings of Royal Society of Edinburgh. [sess. 
It will be observed that the left-hand member involves (1) a skew 
determinant and (2) a bipartite function having the same square 
array as the determinant, and that the right-hand member closely 
resembles the Pfaffian development of § 9. To prove the theorem 
it will suffice to show how the next identity is deduced from the 
last of the preceding three. For shortness’ sake, let us express 
this last in the form 
S 4 + B 4 — E 4 , 
then we have the left-hand member of the next identity, viz., 
S 5 B 5 — S 4 + m 
1 
a l 
a 2 
a 3 
a 4 
ri l ,l 2 
+ B 4 + 
,l z 
n 5 
a 4 
*1 
“ a l 
1 
Pi 
3 
• 
• 
fiz 
k 2 
" a 2 
-Pi 
1 
7i 
72 
• 
• 
72 
^3 
a 3 
-Pi 
“7i 
1 
Si 
• 
• 
Si 
-a 4 
-Pi 
"72 
-«i 
• 
— a 4 — /3g 
— 72 - ^1 
1 
h 
But by § 15 the determinant on the right 
= a 4 2 + (3f + y 2 2 + S 4 2 + |eq a 3 a 4 
2 + |cq a 2 a 4 
2 + | a 2 a 3 a 4 
2 d" | /?i @2 fiz 
CO 
02. 
02. 
/^1 /^3 
7i72 
7i 72 
Si 
72 
*1 
and the bipartite 
— L z* _ ~k j kff 3 & 4 
55 b ^A + h ^A’ 
and, further, 
«W + A>* + y 2 2 + V) - kWMi + h;> hMA 
a Pd,72^i a iP?jYfi 
a 4 |??i 7q 7^ 5 
+ £ 3 I W 7i 2 h b 
+ 7 2 |w h 3 h b 
+ S 4 | 772 72 4 ll b 
h 
Jc 2 Jb 5 
k b 
a 4 
Pz 
72 
K 
Consequently, 
S 5 + B 5 = E 4 + h b k b + a 4 | m \ h b 
K h 
a A 
+ . 
+ Til | cq a 3 a 4 2 + 
@2 @Z 
«1 
