I 
360 
A TENTH MEMOIR ON QUANTICS. 
[693 
arid thence 
a 2 8g = (12ad + 8be) 8b 
+ 46 2 . 8c 
+ 12ci6 . 8d 
+ 2e . 8e 
which is =0, as it should be. 
(12ac2 + 8be) e 
+ 4 6 2 . 3/ 
+ 126 (— bf+ ce) 
+ 2e (— 6ad —10be) 
ade b 2 f bee 
= + 12* +8 
+ 12 
-12 + 12 
-12 - 20, 
380. As already remarked, the leading coefficients of H, I, J, &c., are each of 
them equal to a power of a multiplied by the corresponding covariant h, i, j, ..; hence, 
supposing these leading coefficients, or, what is the same thing, the standard ex 
pressions of the covariants h, i, j, .., v, w to be known, we can calculate the values 
of 8h, 8i, 8j, .., 8v, 8w (= 0, since w is an invariant): and the operation 8, instead 
of being applicable only to the forms containing a, b, c, d, e, f becomes applicable to 
forms containing any of the covariants. The values of 8a, 8b, .., 8v, 8w can, it is 
clear, be expressed in terms of segregates; and this is obviously the proper form: 
but for 8r, 8t, and 8v, for which the segregate forms are fractional, I have given 
also forms with integer coefficients. The entire series is 
Deg-order. 
2.8 
8a = 
0, 
3.5 
8b = 
e, 
3.9 
8c = 
3/ 
4.6 
8d = 
i, 
4.8 
8e = 
— Qad — 106c, 
4.12 
s/ = 
2a 2 6 — 18c 2 , 
5.3 
8g = 
0, 
5.7 
8h = 
26e — 41, 
5.9 
8i = 
— 2ab 2 + 2 ah — 18 cd, 
6.4 
V = 
— n, 
6.6 
81c = 
— 2 aj + 66 s — 9bh + 3c^, 
6.10 
81 = 
— Sabd — 7 6 2 c + 7 eh, 
7.5 
8m = 
— bk —p, 
7.7 
8n = 
4>cj, 
8.4 
80 = 
b 2 g + 66m — 6dj — gh, 
8.8 
8p = 
8abj — 5adg — 106 4 + 156 2 6. — 5bcg + 10cm, 
9.3 
8q = 
0, 
9.5 
8r = 
\ (aq + 66 2 j — 5bdg —jh), = 2b 2 j — 2bdg — 6d 
10.6 
8s = 
— 2 agj + 2b 2 g + 36 2 m + 216$ — 4<bgh + 2 eg 2 — Scq, 
12.4 
8t = 
| (bgm + 46j 2 - 3dgj - hq), =-b 2 q-t hq + 6m 2 , 
13.3 
8u = 
0, ^ 
14.4 
8v = 
1 (— bbgr — lObjo + 5gjk — 12js — 9 nq), = — 6dt — 
19.3 
8w = 
0.