1896 - 97 .] Dr Muir on Bordered Skew Determinant. 
345 
or 
abed 
i l n p 
f 9 h 
j k 
m 
•I / 9 h 
j k 
m . 
It will be seen that the Pfaffian product in the latter identity 
corresponds to Cayley’s 
a/31234- 1234, 
the definition in the one case being 
and in the other 
I / 9 h 
j k 
= M 
gk + h), 
1234 = 12-34 + 13-42 + 14-23. 
3. In the next place I recall the theorem regarding the de- 
velopment of a determinant in terms of co-axial minors having 
zero elements in the diagonal ; that is to say, the theorem of 
which the following is an instance — • 
«i b 2 c 3 d 4 1 = 
• #2 ^3 
+ 
• \ 
\ . h h 
c 2 . c 4 
C 1 C 2 • ^4 
d 2 d 3 
^ • 
+ 
+ a^b 2 c 3 d 4 . 
d z 
For the purpose in view a theorem analogous to this needs to be 
established ; but, as both are cases of a widely general theorem, 
it is better to prove all the cases at once. 
4. Consider any determinant, say the determinant | a 1 5 2 c 3 c? 4 e 5 | , 
and fix the attention on any number of the diagonal elements, say 
the elements c 3 , d 4 , e 5 . It is clear that the terms of the final 
development of the determinant may be separated into four groups, 
viz., 
1. those containing all three of these elements 
2. only two .... 
3. only one .... 
4. none .... 
VOL. XXI. 
3 / 12/97 
none 
z 
