G20 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
f n 
1 Opr/’ 
io£y 
b£y'' 
y 5 
1 
-b 
+ b 3 
- b 3 
+ Ir' 1 ' 
- IT) 
+ b ( 
1 
— 2b 
+ 3b 2 
-4b 3 
+ 5 b 4 ) 
+ ac ( 
1 
— 3b 
+ 6b 3 
-10b 3 ) 
+ a 3 cl ( 
+a :3 e ( 
+a 4 f ( 
1 
— 4b 
1 
+ 10b 3 ) 
— 5b) 
1) 
which is 
1 
0 
ac + 1 
a 3 d 
+ 1 
a 3 e 
+ 1 
a 4 f 
+ 
1 
b 3 -1 
abc 
— 3 
a 3 bd 
— 4 
a 3 be 
— 
5 
b 3 
+ 2 
ab 3 c 
+ G 
a 2 b 3 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 co variants ultimately represented by these letters respec- 
tively, are 
a 
b 
c 
d 
e 
/ 
a + 1 
ae +1 
bd — 4 
c 3 +3 
ac 1 
b 3 -1 
ace + 1 
ad 3 — I 
b 3 e -1 
bed + 2 
c 3 — 1 
a 3 f + 1 
abe -f- 5 
acd + 2 
b 2 d + 8 
be 3 —10 
a 3 d + 1 
abc — 3 
b 3 -2 
satisfying, as they should do, the relation /'= — cdd-\-a~hc — 4c 3 . 
Hence forming the values of orb — 3c 3 and are — 2c/’ it appears that the value of the 
last-mentioned quin tic function is 
(1,0 , c,f ci~b~3c 3 , a 3 e— 2cfX& vY°- 
Writing herein x, y in place of y, and now using a, h, c, d, e, f to denote, not the 
leading coefficients but the covariants themselves (« the original quintic, with £ y 
as facients), we have the form 
A, =(1, 0, c,/, cdb — 3c 3 , a 2 e—2cfjx, yf, 
a new quintic, which is the canonical form in question : the covariants hereof 
(reckoning the quintic itself as a covariant) will he written A, B, C . . . V, W ; and 
will be spoken of as capital covariants. 
