260 A SECOND MEMOIR UPON QUANTICS. [141 
I have worked out the example in detail as a specimen of the most convenient method 
for the actual calculation of more complicated covariants 1 . 
35. The number of terms of the degree 6 and of the weight q is obviously 
equal to the number of ways in which q can be made up as a sum of 6 terms 
with the elements (0, 1, 2, ... m), a number which is equal to the coefficient of xiz e in 
the development of 
1 
(1 — z) (1 — xz) (1 — x 2 z) ... (1 — x m z) ’ 
and the number of the asyzygetic covariants of any particular degree for the quantic 
(*]£#, y) m can therefore be determined by means of this development. In the case of 
a cubic, for example, the function to be developed is 
1 
(1 — z) (1 — xz) (1 — x 2 z) (1 — cc?z) ’ 
which is equal to 
1 + z (1 + x + x 2 + af) + z* (1 -f x + 2x 2 + 2x z + 2x 4 + 2x 5 + x 6 ) + &c.. 
where the coefficients are given by the following table ; on account of the symmetry, 
the series of coefficients for each power of z is continued only to the middle term or 
middle of the series. 
1 
(0) 
1 
1 
(1) 
1 
1 
2 
2 
(2) 
1 
1 
2 
3 
3 
(3) 
1 
1 
2 
3 
4 
4 
5 
(4) 
1 1 2 
3 
4 
5 
6 
6 
(5) 
1 1 2 
3 
4 
5 
7 
7 
8 
8 
(6) 
1 Note added Feb. 7, 1856.—The following method for the calculation of an invariant or of the leading 
coefficient of a covariant, is easily seen to be identical in principle with that given in the text. Write down 
all the terms of the weight next inferior to that of the invariant or leading coefficient, and operate on each 
of these separately with the symbol 
ind. 5.-4- 2 ind. c .- + ... (m- 1) ind. 6' . —, , 
a 6 ' 7 a 
where we are first to multiply by the fraction, rejecting negative powers, and then by the index of the proper 
letter in the term so obtained. Equating the results to zero, we obtain equations between the terms of the 
invariant or leading coefficient, and replacing in these equations each term by its numerical coefficient in the