PEOEESSOE CAYLEY’S NINTH MEMOIE ON QUANTICS. 
45 
Article Nos. 347 to 365. — Sketch of Prof essor Gordan’s proof for the finite Number, 
=23, of the Covariants of a Binary Quintic. 
347. I propose to reproduce the leading points of Professor Gordan’s proof that the 
binary quintic (a, b, c, d, e, ffjfc, y) 5 has a finite system of 23 covariants, viz. a system 
such that every other covariant whatever is a rational and integral function of these 23 
covariants. 
348. Derivation. — Consider for a moment any two binary quantics p, of the same 
or different orders, and which may be either independent quantics, or they may be both 
or one of them covariants, or a covariant, of a binary quantic f. We may form the series 
of derivatives 
(P, 'I')°=P4'» 
0 , ^)'=12 p^ 2 =~d x p . . d t yj/ f 
0, ^) 2 =12 2 (pi^ 2 =B^ . d^—2 . Nd, 
where, however, there is no occasion to use the notation (<p, ^)° (as this is simply the 
product and the succeeding derivatives may (when there is no risk of ambiguity) be 
written more shortly (p-J/), (p-^f, (p^Ok & c - >' in all that follows the word “derivative” 
(Gordon’s Tlebereinanderschiebung ) is to be understood in this special sense. 
349. The degree of the derivative (<p\}/) fr is the sum of the degrees of the constituents 
P, 4/; the order of the derivative is the sum of the orders less 2k; it being understood 
throughout that the word degree refers to the coefficients, and the word order to the 
variables. In speaking generally of the covariants or of all the covariants of a quantic 
f or of the covariants or all the covariants of a given degree or order, we of course 
exclude from consideration covariants linearly connected with other covariants (for 
otherwise the number of terms would be infinite) ; but unless it is expressly so stated, 
we do not carry this out rigorously so as to make the system to consist of asyzygetic 
covariants; viz. it is assumed that the system is complete, but not that it is divested of 
superfluous terms. 
350. Theorem A. — The co variants of a quantic of a given degree m can be all of 
them obtained by derivation from f and the covariants of the next inferior degree 
(m — 1). 
In particular for the degree 1 the only covariant is the quantic f itself; for the de- 
gree 2 the covariants are (ff)°, (fff, ( fff, . . . . : using for a moment (3 to denote each 
of these in succession, the covariants of the third degree are (fff ) 0 , (ftf)\ iff) 2 , ■ ■ • ; and 
so on. 
351. Suppose that the covariants of the second degree (fff, (fff, C fff • ■ • are i n 
this order represented by /3,, /3 2 , (3 3 . . . then the covariants of the third degree written 
in the order 
(ft/)"’ (ft/)- (ft/)" • • • (ft/)", (A/), (A/)" ■ • • (A/)”, (A/)- ft/)" • • • • 
