364 
ON SEMINVARI ANTS. 
364 ON SEMIN V ARI ANTS. [942 
show that df— b 2 d~ is not obtainable by mere derivation from the covariants of degree 
3 of the quartic. 
The quintic and its covariants up to the degree 3 are 
¿=( 
1 
5b 
10 c 
10 d 
5e 
f 
(1, b, c, d, e, f\x, yf; 
e + 1 
/ -1 
bf+l 
bd - 4 
CO 
1 
ce — 4 
c 2 + 3 
cd + 2 
cP + 3 
y)\ 
(e -c 2 , f- cd, bf- d 2 7[x, yf; 
C = (c - b 2 , d — bc, e — c 2 , f— cd, bf— cl 2 , cf— de, df— e 2 ]fx, yf, 
D={ce- c 3 , cf— c 2 d, df— cd 2 , bdf — d 3 \x, y) 3 , 
E = (/— be 2 , bf — c 3 , cf— c 2 d, df — cd 2 , ef— d 3 , f 2 - d 2 e\x, yf, 
F = (d — b 3 , e — b 2 c, f — be 2 , bf — c 3 , cf — c 2 d, df — cd 2 , 
ef— d 3 , f 2 — d 2 e, bf 2 — de 2 , cf 2 — e 3 fx, yf 
Hence all the covariants of the degree 3 are A 3 , AB, AG, D, E, F, where 
AB = (e —c 2 , f—bc 2 , bf—c 3 , cf—c 2 d, bcf-cd 2 , bdf—d 3 , bef - d 2 e, bf 2 — d 2 f\x, yf 
AG = (c- b 2 , d - be, y) n ; 
and the derivatives are 
(A, A 3 f , (A, A 3 ) , &c. weights of leading coefficients are 0, 1, 2, 3, 4, 5 
4, 5, 6, 7, 8, 9 
2, 8, 4, 5, 6, 7 
6, 7, 8, 9 
5, 6, 7, 8, 9, 10 
3, 4, 5, 6, 7, 8.