of Edinburgh, Session 1885—86. 
831 
There is also another alternant expression for the product 
I a f ) 3 1 1 a f ) 3 1 1 a A I > which may be obtained either from (A), or quite 
independently as follows : — 
We have 
! ll“A II a A 1 
h A 
\ (h ) 2 
a x \aj 
1 l 2 P 
1 ^ P 
b,V 
x (ap^) 2 
a l 
a 1 b l 
V 
« 2 2 
a 2 b 2 
v 
a 3 
a 3 & 3 
b 2 
°3 
(B) 
6. Further, the right-hand member of either (A) or (B) is readily 
seen to be 
a* *b 2 b B 
\ 
af , bf ) 3 
a 2 
\ 
a s \b 2 
a 3 
h 
so that by developing this in terms of the elements of the first 
column and their complementary minors, we have the identity 
%^3 1 1 a i^3 II a A I = a i 2 ^2^3 I a 2^3 I — a, '2^f ) 3 I a J ) 3 I L a 3^ ) f > 2 I a f > 2 I * P) 
If now we consider the determinants in this identity as minors of 
the determinant | af) 2 c B | , and apply the Law of Complementaries,* 
we have the conjugate theorem 
^1^2^3 I ^1^2^3 I 2 = ^1 I ^1^3 1 1 ^1^2 1 1 ^2 C 3 I 2 — ^2 I ^1^2 1 1 ^2^3 1 1 ^1 C 3 I 2 
+ b B | a 2 & 3 1 1 a^b B 1 1 
or, after division, by b 1 b 2 b B \ a 2 b 3 || af> B || af) 2 | , 
a A c 3 I 
I ^2 C 3 I 
I \% I 2 
I ^1 C 2 I 
* ‘ P) 
ft 2^3 II a f ) 3 II a ]^2 I ^2^3 I a f ) 3 I ^1^3 I a ~\p3 I ^1^2 I a l^2 I 
Further, if we write a’s for b’s in this result, and vice versa , the 
* Thomas Muir, M.A., Trans. Roy. Soc. Edin., vol. xxx. p. 1. 
