258 
A SECOND MEMOIR UPON QUANTICS. 
[141 
a 3 
d 2 b 
a 2 c 
add 
ade 
abe 
ace 
ade 
ae 2 
be 2 
ce 2 
de 2 
ab 3 
abc 
abd 
acd 
add 
bee 
bde 
ede 
d 2 e 
b 3 
ac 2 
b 2 d 
b 2 e 
bdd 
c 2 e 
dd 
b 2 c 
be 2 
bed 
c 2 d 
cd 2 
c 3 
84. Thus in the case of a cubic {a, b, c, dd$x, y) 3 , the tables show that there will 
be a single invariant of the degree 4. Represent this by 
Aa 2 d 2 
+ Babcd 
+ Gacd 
4- Db 3 d 
+ Fife 2 , 
which is to be operated upon with a3& -f 2bd c + 3cdd- This gives 
+ B 
+ 6^ 
+ 3Z> 
+ 2 B 
+ 2 E 
+ 6(7 
+ 3A 
;+ 4 E 
+ 3 D 
a 2 cd 
ab 2 d 
abc 2 
b 3 c 
i.e. B + §A = 0, SD+2B = 0, &c.; or putting A = 1, we find B = — 0, C— 4, I) = 4, 
E = — 3, and the invariant is 
a?d? 
— 6 abed 
+ 4 ac s 
+ 4 b 3 d 
— 3 b 2 c 2 . 
Again, there is a covariant of the order 3 and the degree 3. The coefficient of a? or 
leading coefficient is 
A a 2 d 
+ B abc 
+ Gb 3 , 
which operated upon with adb + 2bd c + 3c3 d , gives 
+ B 
+i 3 A 
+ 3 G + 2 B 
a 2 c 
ab 2