624 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
380. As already remarked, the leading coefficients of H, I, J, &c., are each of them 
equal to a power of a into the corresponding covariant h, hence, supposing 
these leading coefficients, or, what is the same thing, the standard expressions of the 
covariants h, i, j ... v, w to he known, we can calculate the values of Sh, Si, Sj, . . . 
Sv, Siv (=0, since w is an invariant) : and the operation S 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 S a, Sl>, . . . Sv, Sw can, it is clear, be expressed in 
terms of segregates ; and this is obviously the proper form : but for Sr, St, and Sv, for 
which the segregate forms are fractional, 1 have given also forms with integer co- 
efficients. The entire series is 
Deg-order. 
2.8 S a — 0, 
3.5 Sb = e, 
3.9 Sc = Sf 
4.6 Sd = i, 
4.8 Se = — Sad — lbbc, 
4.12 Sf = 2a~b—18c 2 , 
5.3 Sg = 0, 
5.7 Sh - 2be — il, 
5.9 Si =—2ab*+2ah—l8cd, 
6.4 Sj = — n, 
6.6 Sk =-2aj+6b s -9bh+3cg, 
6.10 SI = — 3 abd~7b' 2 c+7ch, 
7 . 5 Sm = — bk —p, 
7.7 S n = 4cj, 
8.4 So = b 2 g-\-6bm—6dj—gh, 
8.8 Sp = Sabj—badg — 106 4 +156 3 A- — 5bcg-\- 10cm, 
9.3 S q = 0, 
9.5 Sr = f(aq+6b z j—5bdg—jh), =2b z j—2bdg — 6dm, 
10.6 S s =-2agj-\-2b s g-\-3b 2 m-{-2lbdj—4bgh-{-2cg z —3cq, 
12.4 St = \ifgm -\~4bj 2 — Sdgj — hq), = — b 2 q+ hq-\-6m*, 
13.3 Su = 0, 
14.4 Sv = i( — 5bgr—10bjo+5gjk—12js—9nq), = — 6dt — 6mr+nq. 
19.3 Sw = 0. 
It is obvious that for every covariant whatever written in the denumerate form 
(X 0 , I I . . .%x, yY, the second coefficient is equal to the first coefficient operated upon 
by S ; so that the foregoing formulae give, in fact, the second coefficients of the several 
co variants. 
381. It is worth noticing how very much the formulae of Table No. 97 simplify 
