830 
Proceedings of the Royal Society 
Lastly, when the minors involve m - 2 general indices p, q, r, . . . 
x, y , z , since in 
the minor 
| aW . . . g x h y kT | 
| b°c p d q . . . h x k y l z | 
(z-m + 2)', (z-m + 3)' . ..(s-l)' 
(y-m + 2)', (y-w + 3)'. . . (s/-l)' 
(q-m + 2)', (g - m + 3)' . . . (g - 1)' 
(p - m + 2)', (p - m + 3)' . . . (p - 1)' 
with m - 1 like results for the other minors, it will follow as before 
by the application of § 1 that 
P m in | a°lpc q . . Ji y kfl n \ 
m - 2 
= J 2 (afo . . . T). n 
1 a a 2 2 
(Z-m+ 1), (z - m + 2), (z^m + 3) . . . (z-1) 
(2/-W+1), (y-m + 2), (y-m + 3) . , .(y-1) 
,q-m+ 1), (^-m + 2), (^ - m + 3) 1) 
(p - m + 1), (p - m + 2), (p - m + 3) . . . (p - 1) 
5. If it be not already known, there is no difficulty in showing 
that the product of all the determinants of the second order, which 
can be formed from any two rows of elements, can be expressed as 
an alternant. Thus, taking two rows of three elements each, 
we have 
b \ &2 &C 
a 2 
a 3 
a x 
«3 
a, 
«3 
b 2 
h 3 
h 
&3 
h 
— afafs affafctf 2 a 2 bf ? ct-pi 2 a^ 
= K«2 6 3 - a i a f^( a i a A - a 2 a A)( a i a A - a 2 a A) + a i a 2 a 3 
= tf{a 2 af) li a Y af) 2 , a 1 af)f)-{-a 1 a 2 a i \ 
or — } ^2^1^3 5 ^3^1^2) b l b 2 b 3 ' 
(A) 
