[693 
356 
A TENTH 
MEMOIR 
ON QUANTICS. 
this is 
r 
10£V 
iofy 
Sfv* 
V s 
= 1 
-b 
+ b 2 
- b 3 
+ b 4 
- b 5 
+ b ( 
1 
- 2b 
+ 3b 2 
— 4b 3 
+ 5b 4 ) 
+ ac ( 
1 
-3b 
+ 6b 2 
- 10b 3 ) 
+ a 2 d ( 
1 
— 4b 
+ 10b 2 ) 
+ a 3 e ( 
1 
- 5b) 
+ a, 4 f ( 
1), 
which is 
1 
0 
ac + 1 
a 2 d + 1 
a 3 e + 1 
a 4 f 
+ 1 
b 2 - 1 
abc — 3 
a 2 bd - 4 
a 3 be — 5 
b 3 + 2 
ab 2 c + 6 
a 2 b 2 d + 10 
b 4 -3 
ab 3 c —10 
b 5 
+ 4 
The values of a, b, c, d, e, f, considered for a moment as denoting the leading 
coefficients of the several covariants ultimately represented by these letters respect 
ively, are 
a 
b 
c 
d 
e 
/ 
a + 1 
ae + 1 
ac + 1 
ace + 1 
a 2 f + 1 
a 2 d + 1 
bd - 4 
b 2 - 1 
ad 2 - 1 
abe + 5 
abc — 3 
c 2 + 3 
b 2 e - 1 
acd + 2 
1 
w 
bed + 2 
b 2 d + 8 
c 3 - 1 
cr 
o 
1 
H-J 
O 
satisfying, as they should do, the relation 
f- = — a?d + a~bc — 4c 3 . 
Hence forming the values of a?b — 3c 2 and a 2 e — 2cf, it appears that the value of 
the last-mentioned quintic function is 
(1, 0, c, f a 2 b - 3c 2 , a?e - 2c/££, rj) 5 . 
Writing herein x, y in place of r), and now using a, b, c, d, e, f to denote, not 
the leading coefficients but the covariants themselves (a denoting the original quintic, 
with £, y as facients), we have the form 
A=( 1, 0, c,/, a?b — 3c 2 , a 2 c-2cf\x, y) 5 ,