1896 - 97 .] Dr Muir on Bordered Skew Determinant. 
347 
the mode of formation of the right-hand member being (1) to take 
the various combinations of c 3 , d 4 , e 5 , viz., 
1 } C 3 5 d„ e 5 j J 5 ^4^5 ; C 3^4 S 5 i 
(2) to take as co-factors of these their co-factors in the determi- 
nant, only replacing in the latter the elements c 3 , d 4 , e 5 by 0 in 
every case. 
The number of terms in the development is thus seen to be 
1 + C 3)1 + C 3 , 2 + C 3 , 3 
i.e., (1 + l) 3 . 
5. It is very interesting to observe the relation of the various 
cases of this general theorem to one another ; it is useful also, as a 
knowledge of it serves to suggest another mode of proof. Thus, 
taking the simplest possible case — that referred to in § 3 as being 
hitherto well known — viz., 
• $2 
+ 2 ^] 
• b 3 \ h 
&i . b 3 b i b 5 
c 2 . 
C 4 C 5 
Cj c 2 . c 4 c 5 
d-2 d 3 
• ^5 
6^2 ^3 • ^5 
e i e 2 e 3 e 4 * 
e 2 e 3 e 4 • 
(EJ 
+ 
* C 4 
d" 2>A C 3 • ^5 
d 3 . d 5 
e 3 c 4 . 
e 4 . 
and adding the first of the C 55l terms to the term before it, the 
first four of the C 5 , 2 terms to the remaining four of the C 5)1 terms, 
and the first six of the C 5 , 3 terms to the remaining six of the C 5 , 2 
terms we have 
^2 ^3 ^4 ^5 
+ 2^2 
a 3 a 4 a 5 
&i . /; 3 b 4 \ 
Ci . c 4 c 5 
C 1 C 2 • C 4 C 5 
(%2 • ^5 
d Y d 2 d 3 . d ^ 
e i e 2 e 4 • 
6 1 e 2 e 3 ^4 * 
+ 2&2«3 
^ 
+ 2& 2 c 3 (7 4 
^1 % 
• ^5 
e i . 
e i e 4 • 
(E 2 ) 
+ 5 2 c 3 c7 4 e 5 a 1 
