350 Proceedings of Royal Society of Edinburgh. [sess. 
The corresponding identity for the case of the 4th order 
evidently is — 
m l 
ft 
h 3 
CO 
CM 
rH 
II 
’ I ^2 ^2 
- ft 
m 2 
a i 
a 2 
a l a 2 
a l a 2 
— k 2 
— a i 
»h 
Pi 
ft 
Pi 
- /+ 3 
— a 2 
-ft 
rn 4 
+ m 2 /3 1 \m 1 li 2 h 3 
+ m 3 a 2 1 m l 7q h 3 
+ li x h 2 
k 2 k 3 
3 
h\k 2 
Pi 
a 2 
a l 
+ m 2 m 3 h 3 k 3 + m 2 mjifc 2 + rnpnffi 
+ rn x m 2 m 3 m 4 , 
as also may readily he seen on putting 
^4 — a 3 = fi 2 = yi = 0 , >n 3 = 1 
in the identity for the 5th order. 
8. Translating into Cayley’s notation, we find these must be 
written 
a i234 /31234 = 1234 ‘ ^ 1234 + 211 ' ° 234 ' ^ 
+ 21 1 • 22 • 34 • a/334 + 21 1 • 22 • 33 • a4 • /34 
+ a/3 *11 • 22 • 33 • 44 ; 
and 
al23i /3123 = “l 23 '^ 123 
+ 2H -23- a/323 + 2H -22-a3-/33 
+ a/3 • 11 • 22 • 33 . 
The former identity is thus readily seen to he given incorrectly in 
CreUe’s Journ. lv. p. 277, and Collected Math. Papers , iv. p. 73; 
still more incorrectly — indeed, ridiculously so — in Collected Math. 
Papers , ii. p. 203 ; and almost correctly in Crelle’s Journ. 1. p. 300, 
although even here a/311 is put instead of a/3 • 11, to which single 
term it is fortunately equal, as the two other terms cancel each 
other. 
The other identity is quite incorrectly given both in the original 
and in the reprint. 
