« 
378 A TENTH MEMOIR ON QUANTIOS. [693 
covariants as well congregate as segregate of the same deg-order: the congregates 
are distinguished each by two prefixed dots, ..bf &c. No further explanation of the 
arrangement is, I think, required. We see from the table in what manner the 
different covariants present themselves in connexion with the derivation-theory. Thus 
starting with the quintic itself a, we have the two derivatives aci4>, act2, which are 
in fact the covariants of the second degree (deg-orders 2.2 and 2.6 respectively) 
b and c. For the third degree we have the derivatives ab'2, cibl, aco, ac4, ac3, ac2, 
acl : the deg-order of aco is 3.1, and there being no covariants of this deg-order, 
ac5 must, it is clear, vanish identically: ab2 and ac4 are each of them of the 
deg-order 3.3, but for this deg-order we have only the covariant d, and hence ab2 
and ac4 must be each of them a numerical multiple of d; similarly, deg-order 3.5, 
abl and ac3 must be each of them a numerical multiple of e; deg-order 3.7, ac2 
must be a numerical multiple of ab; and deg-order 3.9, acl must be a numerical 
multiple of f: the 7 derivatives, which prima facie might give, each of them, a 
covariant of the third degree, thus give in fact only the 3 covariants d, e, f; and 
in order to show according to the theory of derivations that this is so, it is 
necessary to prove—I o , that aco = 0; 2°, that ac4 and ab2 differ only by a numerical 
factor; 3°, that abl and ac3 differ only by a numerical factor; 4°, that ac2 is a 
numerical multiple of ab: which being so, we have the 3 new covariants. The table 
shows that 
for degrees 2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 
No. of derivatives = V7TT9, 29, 41, 46, 52, 46, 44, 35, 26, 19, 17, 12, 13, 6, 6, 3, 3, 1,1, 0,1 
so that the whole number of derivatives is 429, giving the 22 covariants b, c,..., w. 
While it is very remarkable that (by general reasoning, as already mentioned, and 
with a very small amount of calculation) Gordan should have been able in effect to 
show this, the great excess of the number of derivatives over that of the covariants 
seems a reason why the derivations ought not to be made a basis of the theory. 
It is to be remarked that we may consider derivatives pql, pq2, &c., where p, q 
instead of being simple covariants are powers or products of covariants, but that 
these may be made to depend upon the derivatives formed with the simple covariants. 
(As to this see my paper “ On the Derivatives of Three Binary Quantics,” Quart. 
Math. Journal, t. xv. (1877), pp. 157—168, [681].) 
Table No. 99 (Index Table of Derivatives). 
2 
3 
0 2 4 6 
1 3 5 7 9 
b c 
d e ab f 
aa 4 2 
ab 2 1 
ac 5 4 3 2 1 
2 derivs. 
7 derivs.