405] 
AN EIGHTH MEMOIR ON QUANTICS. 
177 
Article Nos. 308 to 317.—Hermite’s Canonical form of the quintic. 
308. It was remarked that M. Hermite’s investigations are conducted by means 
of a canonical form, viz. if A(=J, =(7) be the quartinvariant of the given quintic 
(a, h, c, d, e, f\x, y) 5 , then he in fact finds (X, Y) linear functions of (x, y) such that 
we have 
(a, h, c, d, e, f\x, yf = {X, y, fk, fk, y\ X'$X, Yf 
(viz. in the transformed form the two mean coefficients are equal ; this is a convenient 
assumption made in order to render the transformation completely definite, rather than 
an absolutely necessary one) ; and where moreover the quadricovariant B of the trans 
formed form is 
= *Jaxy, 
or, what is the same thing, the coefficients (X, y, fk, fk, y, X') of the transformed form 
are connected by the relations 
Xy — 4>y fk + 3k = 0, 'j 
X'y — 4y fk + 3k = 0, : 
XX' — 3yy + 2k = f A, y 
the advantage is a great simplicity in the forms of the several covariants, which 
simplicity arises in a great measure from the existence of the very simple covariant 
d cl 
operator y . (viz. operating therewith on any covariant we obtain again a covariant). 
Cbj\- CL 1 
309. Reversing the order of the several steps, the theory of M. Hermite’s trans 
formation may be established as follows: 
Starting from the quintic 
(a, h, c, d, e, f\x, y)\ 
and considering the quadricovariant thereof 
(a, /3, f§x, y) 2 B 
(fx, /3, 7) are of the degree 2), and also the linear covariant 
Px + Qy J 
((P, Q) are of the degree 5), we have 
(3--4>(xy = A, G 
and moreover 
(a, /3. 7-Py = -C, 
viz. the expression on the left hand, which is of the degree 12, and which is obviously 
an invariant, is = — C, where C is (ut supra) 
C = 9L + JK = -9U+GM. 
c. VI. 
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