272 
A SECOND MEMOIR UPON QUANTICS. 
[141 
then identically, 
<1> 2 - V£T 2 + 4H 3 = 0. 
No. 7. 
a + 1 
b + 4 
c + 6 
(7+4 
e + 1 
vY 
No. 8. 
No. 9. 
ae + 1 
6(2-4 
c 2 +3 
± 4 
ac + 1 
ad + 2 
ae + 1 
be + 2 
6(2 + 1 
b 2 -1 
be - 2 
bd+ 2 
c 2 — 3 
cd- 2 
c 2 -1 
± 1 
± 2 
± 3 
± 2 
± 1 
No. 10. No. 11. 
ace + 1 
ad 2 — 1 
6 2 e - 1 
6cd + 2 
c 3 - 1 
( 
a 2 oi 4- 1 
abc - 3 
b 3 +2 
a 2 e + 1 
abd+ 2 
ac 2 — 9 
6 2 c + 6 
abe + 5 
acd— 15 
b 2 d + 10 
be 2 
ace 
aoJ 2 - 10 
b 2 e + 10 
bed 
c 3 
ade — 5 
bee + 15 
bd 2 - 10 
c 2 d ... 
ae 2 — 1 
bde — 2 
c 2 e + 9 
c<7 2 - 6 
be 2 - 1 
cc?e + 3 
d 3 -2 
± 3 
± 3 
± 9 
± 15 
± 10 
± 15 
± 9 
i B 
No. 12. 
a 3 e 3 
+ 
1 
a 2 bde 2 
— 
12 
a 2 c 2 e 2 
— 
18 
a 2 cd 2 e 
+ 
54 
a 2 d i 
— 
27 
ab 2 ce 2 
+ 
54 
ab 2 d 2 e 
— 
6 
abc 2 de 
— 
180 
abed 3 
+ 
108 
ac^e 
+ 
81 
ac 3 d 2 
— 
54 
Ve 2 
— 
27 
b 3 cde 
+ 
108 
b 3 d 3 
— 
64 
b 2 c 3 e 
— 
54 
b 2 c 2 d 2 
+ 
36 
bc*d 
± 442 
The tables Nos. 7, 8, 9, 10 and 11 are the irreducible covariants of a quartic. 
No. 7 is the quartic itself; No. 8 is the quadrinvariant; No. 9 is the quadricovariant, 
or Hessian; No. 10 is the cubinvariant; and No. 11 is the cubicovariant. The table 
No. 12 is the discriminant. And if we write No. 7 = U, No. 8 = I, No. 9 = H, 
No. 10 = J, No. 11 = <E>, No. 12 = V, 
then the irreducible covariants are connected by the identical equation 
JU 3 - IU 2 H + 4i7 3 + d> 2 = 0, 
and we have 
V = I s - 27 J\