COVARIANTS OF PLANE CURVES, 
1191 
0 0 0 0 
= 2a/ ^ + Sa^ag ^ + (Ga^a^ + Sag^) + {7a^a^ + Icua^) ^ 
da. 
da- 
da. 
d 
+ (Sa^ae + Sa-gag + 4a/) ^ + &c 
0 0 0 
— a.^ 5— -}- 2 ag g“ -}- da^ ~ -f- &c. 
ca„ 
da- 
y 
d d d d 
^-3 = + Saaagp-; + (4a.^a^ + 2 ^ 32 ) + (5a .,+ 5aga/ 
ca- 
0 ff, 
0 
-f- (ba^ttg -j- Qa^a- 3a^“) ^ -I- &c. 
oa.g 
The irreducible matrices of even order are easily seen to be 
1 
'tin — CCn 
= «#6 ~ ^ttgCig + 3av ' 
= a^ag — Gctga^ + ISa^ag — lOcig^ j 
i- 
(2S). 
( 26 ), 
and, generally, when n is even, 
/ I — 2) (n — 3) 
= a,a,i — (n — 2) a^a,i_i +-j—^^ a.^an_^ — &c., 
the last coefficient being half of the value as appearing from this series. 
The matrices of odd orders are found from these by the last form of the process of 
eduction and are 
u. =z a./a-^ — da.,a^a^ + 2a/ 
Urj — a/a,j — da^a^a^ + 2a,a^a^ + 8a/a^ — Gaga/ 
Uq = a./a^ — 7a,a^a^ + — 5a,a^af^ + 12a3^a7 — SOaga^ag + 20a3a5^ 
■ ( 27 ). 
From these it follows that 
Xlglig 
OgMg 
flgWy 
fig Mg 
&C. 
u, 
0 
1 Ou^u^ 
7u,u^ 
2\u,v^ 
lbU,Urj 
■ (28), 
