832 
Proceedings of the Royal Society 
right-hand side changes form and sign, while the left-hand side 
changes sign only ; so that we have a second expression for 
1 aA c s 1 2 . 
I II a i^3 II I 
which two results may be combined in one enunciation as follows : — 
If ( 12 ) denote either 
I ^l C 2 I 
or 
I a i C 2 I 
then 
h A I a A I 
( 23 ) -( 13 ) + ( 12 ) = 
a 4 a 2 
I a-A I ’ 
| a 
1 2 l 3 
| a 2 b 3 | a-Jb 3 1 1 a A 
(i) 
7 . In an exactly analogous way, starting with the two rows of 
elements, 
a x a 2 a 3 a 4 
we can show that 
b 1 b 2 b s & 4 , 
I tt 1^2 1 1 ^1^3 1 1 a i ^4 1 1 a 2p3 1 1 ^A 1 1 | 
= £i(a 2 a 3 aA , a 1 a s a 4 b 2 , a x a 2 af ^ , a l a 2 a 2 b^)-^{a 1 a 2 ayi^ 3 
or^A¥A> aff) 2 \, 3 
and also 
}• • (A,); 
af 
r-dT 1 
(N 
aA 2 
¥ 
a 2 3 
afb 2 
»A ! 
V 
a 3 3 
CO 
rC> 
CM 
CO 
e 
“A 2 
V 
a 4 3 
a A 2 
V 
and therefore 
(B,) 
af 
af x 
V 
a 2 2 
af> 2 
V 
af\b 2 b^ 
a 2 
a A 
V 
afbffz 
a 2 
a A 
¥ 
and thence expanding in terms of the elements of the first column 
and their complementary minors, and using (B), we have the 
identity 
j af 2 || a A 1 1 a A 1| of 2 & 3 1 1 a A 1 1 a A | 
= 0^6 A& 4 I a A 1 1 a A \ | a A | - afb A& 4 | a A | ] a A 1 1 a 3 & 4 | 
+ a 3 A&A | a A 1 1 aA j| a 2 & 4 1 — | a A j | a A 1 1 ^A 1 • • • (Qi) 
