344 Proceedings of Royal Society of Edinburgh. [sess. 
The statement of the identity for the case of the determinant of 
the 4th order ought to be a help in ascertaining the law of the 
development, but such is not the case. With the two identities 
before him, no reader would find it possible, without investigation, 
to write out the identity for the case of a determinant of higher 
order. 
So far as I am aware, there is no other literature on the subject, 
which seems indeed to have been entirely lost sight of by mathe- 
maticians, although a very special case of the theorem has received 
considerable attention. 
Having recently been led by a straightforward process of deduc- 
tion to what turns out to be the general theorem which includes 
the two instances intended to be stated by Cayley, I am in a 
position, not only to remove the uncertainty above referred to, but 
also to supply the much-needed proof. 
2. First of all, I recall the very special case which has already 
been proved by several investigators,* and which is best known in 
the form : — Every bordered zero-axial skew determinant is expressible 
as the product of two Pfafflans. This is necessary as a lemma to 
what follows. The mode of formation of the two Pfaffians is 
readily apparent from two examples. 
(a) Order even — 
a 
b 
c 
d 
e 
= | a b c 
d e 
. 
1 “ P 
r 
8 
e 
- a 
f 
9 
h 
i 
f 9 
h i 
/ 
9 
h 
i 
1 
1 
9 
j 
k 
l 
j 
k l 
3 
k 
l 
-y -a 
-/ 
• 
m 
n 
m n 
m 
n 
5 -h 
-k 
- m 
9 
P 
P 
P 
- e - i 
-l 
- n 
~P 
• 
(b) Order odd- 
e 
a 
b 
c 
d 
= | a b c 
d 
e 
• 1/ 
9 
h 
- i 
. 
f 
9 
h 
f 9 
h - 
- i 
3 
k 
-l 
-/ 
© 
9 
k 
j 
k - 
-l 
TO 
i 
— n 
-9 
• 
m 
m - 
- n 
~P 
-h 
-k 
- m 
• 
~P 
* Quart. Journ. of Math., xviii. pp. 46-49. 
