of Edinburgh, Session 1885-86. 
833 
With the help of the Law of Complementaries we should then 
obtain the conjugate theorem 
I ^l C 2 1 1 ^1 C 3 1 1 ^1 C 4 1 1 ^2 C 3 i I ^2 C 4 I i ^3 C 4 I ! a i^2 C 3^4 i 3 
= | b 1 C 2 || 6/ 3 116^ || df>2, C ^ II ^1^2 C 4 II a i^2 G d II ^2 C 3^4 I 3 
— | 6/2 | . . | af 2 c 3 | . . | b^ 3 d 4 | 3 + | 6/g 1 . . | af 2 c 3 | • • | ^i c 2^4 i 3 
— | 6/4 1 1 &2 C 4 1 1 ^3 C 4 1 1 a X^2 C 4 1 1 a l^3 C 4 I l^2^3 C 4 I I ^1 C 2^3 I 3 ’ 
or, after division by 
I Vf 1 1 Vs | ... | b 3 c 4 1 1 a 2 b 3 c 4 1 1 a x b 3 c 4 1 1 af 2 c 4 | | af 2 c 3 | , 
I V2V4 1 3 
I a 2^3 C 4 1 1 ®1 V4 1 1 a l^2 C 4 1 1 a l Vs I 
= I hc- 3 d 4 I 3 1 VA1 8 
I Vs 1 1 6/ 4 1 1 b 3 c 4 1 1 a 2 6/ 4 1 | 6/ 3 1 1 6/ 4 1 1 6/4 1 1 af 3 c 4 | 
_l I \c 2 d 4 | 3 I b x c 2 d 3 | 3 (d ) 
I V2 1 1 V4 1 1 V4 i I °h V4 I I V2 1 1 V.3 1 1 V3 1 1 a l V3 I 
As before, we observe that, if in this result we write a’s for 6’s and 
vice versa , and again if in the result so obtained we write 6’s for c’s 
and vice versa , in each case the right-hand side changes form, while 
the left hand side remains unaltered, so that we obtain two addi- 
tional expressions for this side; hence — 
If (123) denote 
I V2^s 1 3 [ 
I Vs I p m ’ I a 
then 
(234) - ( L34) + (124) - (123) = ^ 3 ■ • ■ (II.) 
m 
where in each expression P w denotes the product of the comple- 
mentary minors of the elements of the last column of the deter- 
minant in the numerator. 
a \ C 2^2> I 
1V3 I 
or 
I a f 2 d 3 | 3 
I a i V3 I P ? 
VOL. XIII. 
3 
