414 CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEYENTHIC. [696 
is as follows: 
Ind. x 
0 2 4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
i i i 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2 
2 
3 
3 
4 
4 
5 
5 
(l-« 4 )- 1 
11 2 
2 
3 
3 
4 
4 
5 
5 
6 
6 
1 
1 
3 
3 
6 
6 
10 
10 
15 
15 
(l-« 4 )- 1 
1 1 3 
3 
6 
6 
10 
10 
15 
15 
21 
21 
1 
1 
3 
4 
7 
9 
14 
17 
24 
(1 — if 6 ) -1 
1 1 3 
4 
7 
9 
14 
17 
24 
29 
38 
45 
1 
1 
3 
5 
8 
12 
19 
25 
36 
1 1 3 
5 
8 
12 
19 
25 
36 
48 
63 
81 
1 
1 
3 
5 
9 
13 
22 
30 
(l-« 8 )- 1 
1 1 3 
5 
9 
13 
22 
30 
45 
61 
85 
111 
the alternate lines giving the developments of the functions 
(1 - x 2 )~\ (1 - w 2 )- 1 (1 - a 1 )- 1 , (1 - x 2 )- 1 (1 - x 1 )- 2 , ..., 
which are the products of the «-functions down to any particular line. And in like 
manner we have the expansions of the other functions F, G, H respectively. I give 
first the expansions of E, F, G, H; next the calculation of the X’s; then the cal 
culation of the F’s: and from these the Z’s up to Z m , or coefficients of the powers 
a°, a 1 , ..., a 18 in the numerator of the N.G.F. are at once found; and the coefficients 
of the remaining powers a 19 , ..., a 36 are then deduced, as already mentioned. 
Writing in the formula x = 0, we have, for the numerator of the N.G.F. of the 
invariants, the expression 
1 - a 6 + 2a 8 - a 10 + 5a 12 + 2a 14 + 6a 16 + 2a 18 + 5 a 20 - a 22 + 2« 24 - a 28 + a 32 , 
agreeing with a result in my “ Second Memoir on Quantics,” Phil. Trans., t. cxlvi. 
(1856), [Number 141, vol. n. in this Collection, p. 266]; this, then, was a known result, 
and it affords a verification, not only of the terms in x°, but also of those in « 14 . Thus, 
in calculating the foregoing expression of the numerator, we obtain Z 4 — (2« 4 + « 8 + « 14 ), 
viz. the term is 
a 4 (2« 4 + ofi + « 14 ), 
and we thence have the corresponding term a 32 (1 + « 6 + 2« 10 ), which, when x — 0, 
becomes = a 32 , a term of the numerator for the invariants: and the term l« 14 of Z±