104 Proceedings of Royal Society of Edinburgh. [jan. 6, 
Thus taking for A and B the determinants 
I af^cgd,^ I , I af^d^e^ | , 
and forming from them the set of six determinants 
^2 
^3 
«4 
^1 
«2 
CO 
«4 
^2 
«3 
a^ 
h 
h 
h 
^8 
h 
h 
h, 
h 
% 
^6 
^8 
^1 
^2 
^3 
^4 
«5 
^6 
^8 
^5 
^6 
ds 
j 
d^ 
d& 
dr^ 
C?8 
j 
d. 
C?2 
<^3 
d. 
«6 
a^j 
«8 
«5 
Uq 
a>j 
«8 
h 
h 
h 
h 
^4 
h 
b, 
^8 
^1 
^2 
% 
^4 
^5 
^6 
^8 
^2 
^3 
^4 
C?6 
dfj 
ds 
j 
d. 
(^2 
C?3 
d^ 
j 
d. 
^2 
^3 
d. 
by repeatedly choosing two rows from A and two from B, we then 
expand each of the six in terms of minors of the second order and 
their complementary minors, the minors being formed in every case 
from the rows originally taken from A. There will be six terms in 
each of the six expansions ; so that if we write the second expansion 
under the first, the third under the second, and so on, we shall 
have a collection of thirty-six terms which may be viewed as 
arranged in six columns as well as in six rows. Expressing the 
sum of the terms in each column as a determinant, we find for the 
total 
\af^c^d^\ - 1 - + Vhh^id^\ 
-t- \af>.2^^d^ -p \af>2^'^d^ + Wf^cyi ^ , 
in accordance with the theorem. 
9. If A and B be the alternants 
the identity becomes 
{aP + + a^d^ -P -l- h^d^ + c^df\a%'^cH^\ 
= -p \a%^+hH^+^\ -P -p 
! + \a^h'^rPH^\ + , 
or 
= a( 0 ,l,s -P 2 ,s -P 3 ) -p a( 0 ,s -P 1 , 2 ,s + 3 ) -P a( 0 ,s + l,s -h 2 , 3 ) 
-p a( 5 ,l, 2 ,s -p 2 ) -p a(s,l,S -P 2,3) -p a(s,S -P 1,2,3), 
which we know on other grounds to be true. 
