48 
PROFESSOR CAYLEY’S NINTH MEMOIR ON QTJANTICS. 
358. The proof that for a form f of the order n the number of covariants is finite, 
depends on the assumption that the number is finite for a form/ 7 of the next inferior 
order n— 1 : this being so, the number of the special covariants of/ will be finite ; say 
these are A 1? A 2 , A 3 . . . (/is itself one of the series, but we may separate it, and speak 
of the form / and its special covariants) : the forms W are functions of the special cova- 
riants, and hence every covariant whatever of/ is of the form F(A)-j-P x ; but it requires 
still a long investigation to pass from this to the theorem of the existence of a finite 
number of forms V such that every covariant whatever is F(V). I pass this over, and 
reproduce only the investigation for the case of the quintic. 
359. Starting from the assumed system of forms, 
/> *=(//) 4 »i=(/0 3 s «=(/?> P=(&)\ *■=(. pi)\ y=(ra), 
(fp)> (fp)> (f T )> 0)> 
(fi)> 0 ‘)> (jfy («)» 
(?‘a), (iy), {H)\ ((*«), «), («>)*, (/a), y), 
say, the 23 forms U, it is to be shown that every other covariant whatever of the quintic 
is of the form F(U). 
The special covariants are/, <p, (/p), i,j, which are forms U ; the only form ^ is i, so 
that instead of P x writing P i5 every co variant whatever of / is 
= F(U)+P i ; 
so that it remains to show that every form P f is F(U) ; or, what is the same thing, that 
if H be any form F(U) whatever, then that (PIQ and (H if are each of them F(U). 
360. In order to show that every covariant of a degree not exceeding m is F(U), it will 
be sufficient to show that the several forms (TIQ and (HQ 2 of a degree not exceeding m 
are each of them F(U) ; and if for this purpose we assume that it is shown that every 
covariant of a degree not exceeding m — 1 is F(U), then in regard to the forms (ITQ and 
(HQ 2 of the degree m, it will be sufficient to show that any such form is a function of 
covariants of a degree inferior to m. 
361. First for the form (HQ: we have (PQ, Q=P(QQ + Q(PQ; and hence we see 
that (HQ will be F(U) if only (UQ is always F(U). 
In forming the derivative of i with the several covariants U, we may omit i itself, and 
also the four invariants (iif, (irf, ((ice), a), ((ice), y), since in each of these cases the 
derivative is =0. We have therefore to consider the derivative of i with 
f, <{>J> (/<?)> (fp), (fr), (jr), (// (pi)> («)> (*«), (iy\ 
respectively : the first seven of these are each of them U; the remaining eleven are each 
of them of the form ((PQ), Q. Now ((PQ), Q is a linear function of P(QQ 2 , Q(PQ 2 , and 
QPQ) 2 , that is ((PQ), Q is a function of covariants of a lower degree than itself. 
362. Next for the form (HQ 2 , we have (PQ, Q 2 , a linear function of P(QQ 2 , Q(PQ 2 ; 
QPQ) 2 ; and we hence see that (HQ 2 will he F(U) if only (UQ 2 is always F(U). 
