430 
[698 
698. 
ON A THEOREM RELATING TO CO VARIANTS. 
[From the Journal für die reine und angewandte Mathematik (Crelle), t. lxxxvii. (1878), 
pp. 82, 83.] 
The theorem given by Prof. Sylvester, Crelle, vol. lxxxv., p. 109, may be stated as 
follows: If for a binary quantic of the order i in the variables, we consider the 
whole system of covariants of the degree j in the coefficients, then 
10 (k + 1) = 
R (i + j) 
noon or 
where 6 denotes the number of asyzygetic covariants of the order 6 in the variables, 
the values of 6 being ij, ij - 2, ij — 4,..., 1 or 0, according as ij is odd or even. 
In the case of the binary quintic {a, ..Jfx, y) 5 , (i — 5), we have a series of 
verifications in the Table 88 of my “ Ninth Memoir on Quantics,” Phil. Trans, vol. CLXI. 
(1871), [462]: viz. writing the small letters a, h, c, ..., u, v, w (instead of the capitals 
A, B, etc.) to denote the covariants of the quintic, a, the quintic itself, degree 1, 
order 5, or as I express it, deg-order 1.5: b, the covariant deg-order 2.2, etc., the 
whole series of deg-orders being 
a, 
b, c, 
d, 
e, f, 
9> 
h, 
i f 
k, l, 
1.5, 
2.2, 2.6, 
3.3, 
3.5, 3.9, 
4.0, 
4.4, 
4.6, 5.1, 
5.3, 5.7, 
m, 
n, 0, 
q, r, 
s, 
t, 
u, V, 
w, 
6.2, 
6.4, 7.1, 
7.5, 
8.0, 8.2, 
9.3, 
11.1, 
12.0, 13. 
1, 18.0, 
then the table shows for each deg-order, the several covariants of that deg-order, and