693] 
A TENTH MEMOIR ON QUANTICS. 
357 
a new quintic, which is the canonical form in question : the covariants hereof 
(reckoning the quintic itself as a covariant) will be written A, B, C,..,V, W, and 
will be spoken of as capital covariants. 
376. The fundamental property is : Every capital covariant, say I, has for its 
leading coefficient the corresponding covariant i multiplied by a power of a : and 
this follows as an immediate consequence of the foregoing genesis of A. The 
covariant i of the form 
i(a, b, c, d, e, f$£ y) 5 
d 
has a leading coefficient 
= ^ (a 2 cf — a 2 de + &c.), 
a 4 v 7 
which, when a, b, c, d, e, f .., i denote leading coefficients, is = i multiplied by a power 
of a: and upon substituting for the quintic the linear transformation thereof 
(1, 0, c, / a?b — 3c 2 , a 2 e-2c/$£, y) 5 , 
(observing that, in the transformation |, y into £ — by, ay, the determinant of sub 
stitution is = a), the value is still = i multiplied by a power of a ; or using the 
relation a = a, say the value is —i multiplied by a power of a. Now the covariant 
i is the same function of the covariants a, b, c, d, e, f that the leading coefficient 
i is of the leading coefficients a, b, c, d, e, /; hence, the italic letters now denoting 
covariants, the leading coefficient still is =i multiplied by a power of a: which 
is the above-mentioned theorem. 
377. To show how the transformation is carried out, consider, for example, the covariant 
B. This is obtained from the corresponding covariant of (a, b, c, d, e, f$£, y) 5 , that is, 
ae 
1 
af 
1 
bf 
1 
bd 
- 4 
be - 
- 3 
ce 
- 4 
c 2 
+ 1 
cd + 1 
d 2 
+ 3 
by changing the variables, and for the coefficients 
a, b, c, d, e, f 
writing 
1, 0, c, /, a-b — 3c 2 , a 2 e — 2c/; 
thus the coefficients are 
First. 
1 (a 2 6 - 3c 2 ) 
+ 3c 2 
= ad) 
Second. 
1 (u 2 e — 2c/) 
+ 2 of 
= a?e 
Third. 
— 4 c (a-b — 3c 2 ) 
+ 3/ 2 
= — 4a 2 6c + 12c 3 
+ 3 (— a 3 d + a 2 bc — 4c 3 ) 
= a 2 (— Sad — be) ;