122 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
No. 7- 
+ 1 a 
+4 b 
+ 6 c 
+ 4 d 
+ 1 e 
yY 
No. 8. 
No. 9. 
+ 1 ae 
-4 hd 
-j- 3 c“ 
+ 1 ae 
+ 2 ad 
+ \ ae 
+ 2 be 
1 bd 
-1 b‘ 
— 2 be 
+ 2 bd 
-3 
—2 cd 
-1 
4 
No. 10. No. 11. 
+1 
ace 
+ 1 
a^d 
+ 1 
9 
a“e 
+ 5 
abe 
ace 
- 5 
ade 
— 1 
ae^' 
— 1 be^ 
— 1 
ad~ 
f 
-3 
abe 
+ 2 
abd 
-15 
acd 
-10 
ad^ 
+ 15 
bee 
— 2 
bde 
+ 3 ede 
— 1 
b^e 
\ 
+ 2 
-9 
ac“ 
+ 10 
Vd 
+ 10 
Ve 
-10 
bd^ 
+ 9 
c‘e 
-2 d^ 
+ 2 
bed 
+ 6 
bc^ 
bed 
cV 
-6 
edr 
— 1 
c® 
No. 12. 
+ 
1 
ede^ 
+ 81 
ac*e 
— 
12 
d^bde“ 
- 54 
ac^d? 
— 
18 
cre^e^ 
- 27 
+ 
54 
d^cd'^e 
+ 108 
b^ede 
— 
27 
ard'^ 
- 64 
b^d^ 
+ 
54 
afj^ce^ 
— 54 
Vc^e 
—• 
6 
ab‘d^e 
+ 36 
bW 
— 
180 
abdde 
bc^d 
+ 
108 
abed^ 
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 cubico variant. The table 
No. 12 is the discriminant. And if we write 
No. 7=U, 
No. 8=1, 
No. 9 = H, 
No. 10=J, 
No. 11=0, 
No. 12=V, 
then the irreducible covariants are connected by the identical equation 
JU^-IU^H+4H^+O^=0, 
V=P— 27J". 
and we have 
