834 
Proceedings of the Poyal Society 
In like manner it can be shown that, 
If (1234) denote 
1 VAfrl 4 1 <h C 2 d B e i I 4 ? 1 «iM 3 g 4 l 4 
I ^i^2 c 3 ^4 I I d^ I h m | af 2 cyl± | P„ 
or 
then will 
| | P m 
(2345) - (1345) + (1245) - (1235) + (1234) 
I a A C A e b I 4 
(III.) 
8. It is thus evident that the theorem is a general one, so that 
starting with the two rows of m elements each, 
b 1 b 2 b s . . b m , 
we can show that the product of all the determinants of the second 
order which can he formed from them, viz. 
\ a A\\ a A\\ a A\- • • • \ a m-A\, 
which consists of \m{m - 1) factors, 
= £{a 2 a z . . a m b lt a Y a 3 . . a m b 2 , . . . , a Y a 2 . . a m _ x b m ) + (a ± a 2 a 3 . . a m ) x 
or 
( _ i )£m(m i)g 2 (af>f>^..b m) "b m > • ••> ei^f> 2 ..b m _i^ — (ff 2 b^...b m ) :i 
(A 2 ) 
where the index x = ^(m - 1 )(m - 2); and also 
i—l m 
ap~% 2 . . . bf- 1 
a 2 m ~ 2 b 2 a 2 m ~ 3 b 2 . . . bp- 1 
1 cC~ 2 b m aZ~ 3 bJ . . . IZ~ X 
• (B 2 ) 
and therefore 
a i m ~\h - bm, af 2 , «i m . 
a 2 m >b A ■ ■ - K, "i"-' 1 . , . 
• V 
7, »»-a 
. U 2 
\b 2 . , . , C 2 , C 3 b m , . . . hi 
