1896-97.] Dr Muir on Bordered Skew Determinant. 
357 
or, for compactness’ sake, 
CL i CL^CLq 
a iftyi 
In the next place, the product | a l b 2 c 3 | • | a 1 p 2 y 3 \ • I aq?/ 2 z 3 I 
* has elements of the form 
CL' i CLcXLo 
X, 1 2 3 + 
a \P\i\ 
Vl 
a \ a 2 a S | a } a 2 a S 
a 2fi i fl2 1 a sPs7 3 
or cq 
«2 
a s 
j 
cq 
ft 
7i 
a 2 
@2 
72 
Vi 
a 3 
Ps 
7s 
z i 
the final expansion of the latter expression being easily obtainable 
on remembering that there is a term corresponding to every ele- 
ment in the square array, and that this term is the product of that 
element, P 3 say, and the two outside elements a 2 and z 1 standing 
in the same column or row with it. 
Further than these two cases it is not necessary at present to go. 
The succeeding expressions of the same kind will he found in a 
paper published in Transactions of the Society, where also an ex- 
position of their properties is given.* 
17. 
Now, the first three instances of 
our new theorem are- 
/ft 7ft 
1 2 
7ft 7>._ 
m 
1 cq 
+ 1 Ct| 
Aq = m + 
'i 2 
Zq k 0 
+ cq | 7q /? 2 
— oq 1 
-a, 1 
k 2 
1 2 
k>i /ft 
a i 
7ft 
ho h 0 
1 
a i 
a 2 
+ 
1 
1 
'2 3 
a l a 2 
7q = 
+ 
/q/^2^3 
Zft Zft/ft 
+ Sajm 7q /ft 
“ a i 
1 
ft 
-oq 
l ft 
K 
12 3 
^1 ^2 
- a 2 
-ft 
1 
-a 2 
-ft l 
k 3 
«] 
+ 
/ft /ft /ft 
1 
a i 
a 2 
a 3 
1 oq a 2 
a 3 
7 1 /ft 7ft /ft /ft , 
k, = m + - 1 - ft ft 4 2, + 
k, kJtok. 
~ a i 
1 
ft 
P 2 
— oq 1 
ft 
P 2 
k% 
± z o * 
- a 2 
-ft 
1 
7i 
~ a 2 — P] 
1 
7i 
k 3 
-a 3 
~ P 2 
“ 7i 
1 
- a 3 - ft - 7l 
1 
ft 
+ m|oq a 2 a 3 
2 
Pi @2 
7i • 
* “ On Bipartite Functions,” Trans. R.S.E., xxxii. pp. 461-481. 
