214 Proceedings of the Royal Society of Edinburgh. [Sess. 
where the determinant on the right differs only in sign from the last 
determinant in (III.). By combination of (III.) and (IV.) there thus follows 
I A 1 A 2 B 2 B 3 C 3 C 4 D 4 L\ I = - A 2 I a Y a 2 b 2 b 3 c 3 c 4 d 4 d x | . (V.) 
(5) If before operating on | A^ , B 2 B 3 , C 3 C 4 , D 4 D 4 i we had made its 
2nd, 3rd, 4th columns the 3rd, 4th, 2nd respectively, we should have 
obtained on the right of (III.) 
a 3^1 
«iA 3 
a 3 A, 
^3^1 
t^jBg 
i>A 
c 3 Ci 
c-j C 3 
c s a 
^3 D l 
,l 3 D. 
There thus results the curious identity, 
a 2 A 4 
«4 A 4 
« 4 a. 
& 2 B 2 
b 2 B 4 
6 4 B ; 
« 2 c 2 
c 2 C 4 
c i C 4 
c 4 C, 
d 2 D 4 
d t D 4 
dp, 
a 1 A 1 
%Ai 
a \A$ 
a 3 A. 
bp 
b sA 
&iB 3 
63B; 
Ofil 
c 3 Ci 
C]C 3 
c 3 C. 
dp. 
d^D l 
d\ D3 
d^D, 
(VI.) 
or, in words, the determinant obtained from | A X A 2 , B 2 B 3 , C 3 C 4 , D,^ | by 
changing A 2 , B 2 , C 2 , D 2 , A 4 , B 4 , C 4 , D 4 into a 1 ,b 1 , c 1} d v a 3 , b 3 , c 3 , d 3 , 
differs only in sign from that obtained by changing A x , Bj , , D x , 
A 3 , B 3 , C 3 , D 3 into a 2 ,b 2 , c 2 , d 2 , a 4 ,b 4 , c 4 , d 4 . 
It will be found to be a direct consequence of the fact that the inter- 
changes 
p a i > 
* 1 . 
c i y 
<*l\ 
/ a 3 J 
CO 
C 3 > 
^s\ 
L ^2 > 
b 2 , 
c 2 > 
dej , 
^ 4 ) 
C 4 > 
dj 
entail the interchanges 
71 Aj, Bj , C j, Da t>A 3 , 1*3 > ^V\ 
U 2 ,B 2 ,C 2 ,dJ 5 L A 4 , B 4 , C 4 , D 4 / , 
and that the said interchanges made in (III.) do not alter A and alter 
| A 4 A 2 B 2 B 3 C 3 C 4 D 4 D x I merely in sign. 
(6) Denoting the determinants of (VI.) by M and N, we have 
1ST = 
A 4 a 3 
A z a 3 
Bj | ct 3 b 1 1 
Ah 
B 3 | a 3 \ 1 
^ 3^3 
Ci | CL^Pl 1 
CiC 3 
C 3 1 a 3 C \ 
^3 C 3 
Di|« 3 ^iI 
D \d 3 
D 3 1 a 3 d x | 
D 3 ^3 
2 
