272 
[761 
761. 
ON THE THEOREM OF THE FINITE NUMBER OF THE 
COVARIANTS OF A BINARY QUANTIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881), 
pp. 137—147.] 
Goedan’s proof, the only one hitherto given, is based upon the theory of derivatives 
(Uebereinanderschiebungen). It is shown that the irreducible covariants of the binary 
quantic f are included in the series 
(f /)*> - if h), (/, h) 2 ,... 
of the derivatives of the quantic upon itself or upon some other covariant, and that 
the number of the irreducible covariants thus obtained is finite. And not only so, 
but for the quintic and the sextic the complete systems were formed, and the numbers 
shown to be = 23 and 26 respectively. 
It would seem that there ought to be a more simple proof based upon the con 
sideration of the fundamental covariants: for the cubic (a, h, c, dfgx, y) s , these are 
the cubic itself (a,...\x, y) 3 , the Hessian (ac — b 2 , y) 2 , and the cubicovariant 
(a 2 d — Sabc + 26 s , y) 3 ; and so in general for the quantic y) n , we have a 
series of fundamental covariants the leading coefficients whereof are the seminvariants 
a, ac — If, a 2 d — Sabc + 2b 3 , a 3 e — 4a 2 bd + 6ab 2 c — 3¥, &c. 
It is known that every covariant can be expressed as a rational function of these, or 
more precisely that every covariant multiplied by a positive integral power of the 
quantic itself can be expressed as a rational and integral function of the fundamental 
covariants, and we may for the covariants substitute their leading coefficients, or say 
the seminvariants; hence, every seminvariant is a rational function of the fundamental 
seminvariants, and more precisely, every seminvariant multiplied by a positive integral