693] 
A TENTH MEMOIR ON QUANTICS. 
377 
Derivatives. Art. Nos. 382 to 384, and Tables Nos. 99 and 100. 
382. I call to mind that any two covariants a, 6, the same or different, give 
rise to a set of derivatives (a, ft) 1 , (a, 6) 2 , (a, 6) 3 , &c., or, as I propose to write them, 
«61, «62, «63, &c., viz.: 
«61 = d x a ,d y b — d y a. d x b, 
«62 = d x a. d y 2 b — 2d x d y a. d x d y 6 + d y 2 a. d x 2 b, 
«63 = d x a. d y 3 b — 3 d x d y a. d x d y b + ‘¿d x d y a. d x d y b — d y a. d x b, 
&c.; 
or, as these are symbolically written, 
a61 = 12«j6 2 , «62 = 12 2 a 1 6 3 , «63 = 12 3 «!6 2 , &c. ; 
where 
19-fc & _ d d d_ d 
u-frv*-s*h, -¿¡r d j 2 -¿¡T d -> 
the differentiations applying to the a x and the 
but the suffixes being ultimately omitted : hence if 6 be the index of derivation, the 
derivative is thus a linear function of the differential coefficients of the order 6 of 
the two covariants « and 6 respectively : and we have the general property that any 
such derivative, if not identically vanishing, is a covariant. If the « and the 6 are 
one and the same covariant, then obviously every odd derivative is = 0 ; so that in 
this case the only derivatives to be considered are the even derivatives ««2, ««4, &c. : 
moreover, if the index of derivation 6 exceeds the order of either of the component 
covariants, then also the derivative is = 0 : in particular, neither of the covariants 
must be an invariant. The degree of the derivative is evidently equal to the sum 
of the degrees of the component covariants ; the order is equal to the sum of the 
orders less twice the index of derivation. 
d 
dx. 2 
dy 2 
applying to the 6 2 , 
383. It was by means of the theory of derivatives that Gordan proved (for a 
binary quantic of any order) that the number of covariants was finite, and, in the 
particular case of the quintic, established the system of the 23 covariants. Starting 
from the quantic itself «, then the system of derivatives ««2, ««4, &c., must include 
among itself all the covariants of the second degree, and if the entire system of these 
is, suppose, 6, c, &c., then the derivatives «61, a62, &c., «cl, ac2, &c., must include 
among them all the covariants of the third degree, and so on for the higher degrees; 
and in this way, limiting by general reasoning the number of the independent 
covariants of each degree obtained by the successive steps, the foregoing conclusion 
is arrived at. But returning to the quintic, and supposing the system of the 23 
covariants established, then knowing the deg-order of a derivative we know that it 
must be a linear function of the segregates of that deg-order; and we thus confirm, 
d 'posteriori, the results of the derivation theory. I annex the following Table No. 99, 
showing all the derivatives which present themselves, and for each of them the 
c. x. 48