828 Proceedings of the Royal Society 
with like results for the other four minors, we have 
P m in | a% l c q d r e n \ 
= P m in | a Q b 1 c 2 d^e n | multiplied by the product of the cofactor of 
£*( bcde ) last written and four like expressions with b, c, d } e re- 
spectively in the first row instead of a ; and thus we have 
P 
m 
Lastly, in 
1 
£$(abcde) . II 
(r - 4) 
(2-4) 
a a 2 
(r-3) (r- 2) . 
(2-3) (q- 2) 
| a°b p c q d r e n | , 
since the minor 
| b°c p d q e r | = @(jbcde) 
(r- 3)' 
(2-3)' 
(p-sy 
(r - 2)' (r-l)' 
(2-2)' (?-l)' 
(p-2)' (p-1)' 
1 
= t\bcde) 
(r-4) 
(2-4) 
(P~ 4) 
a a 2 
(r-3) (r-2) 
(2-3) (2-2) 
(P-3) (P-2) 
a 3 
(r-l) 
(2-1) 
(P" 1 ) 
b y§ i. 
with like results for the other four minors, we have 
P m in | a°b p c q d r e n | 
= P m in j a°b l c 2 d 3 e n | multiplied by the product of the cofactor of 
£*(bcde) last written and four like expressions with b , c, d, e re- 
spectively in the first row instead of a ; and thus 
P m = £f (abode) . II 
(r-4) (r-3) (r- 2) (r-l) 
(2-4) (2-3) (2-2) (2-1) 
(P-4) (p-3) ( 2 ? -2) (p-1) 
Generally, in the case of m letters a, b, c, ... h, Jc, Z, since in 
| a°b l c 2 . . . h m ~ 2 Pl n | , 
the minor 
| bWd 2 . . . k m ~H z ! = $(bcd . . . l)h ' z _ m+2 , 
which by § 1 
£*(M ; . . I 
1 
( 2 -m-i-l) (z - m + 2) 
