622 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
disregarding these exterior factors, the leading coefficients for B, C, D, E, F are 
b, c, ad, e,f respectively ; that for G is l2abd-\-4d> 2 c-\-e z , which must be —g into a 
power of a, and (in Table 07) is given as =a°g, similarly that for IT is 6acd-\-4bc~-\-ef, 
which must be =h into a power of a, and is given as =a' 2 h : and so in the other cases. 
The index of a is at once obtained by means of the deg-order, which is in each case 
inserted at the foot of the coefficient. 
For A, B, C, E, F there is no jDower of a as an interior factor, and for the invariants 
G, Q, U we may imagine the interior factor thrown together with the exterior factor, 
(G =a 6 g, &c.): whence disregarding the exterior factors, we may say that for A, B, 
C, E, F, G, Q, U the standard forms are also “ divided ” forms. But take any other 
covariant — for instance, D : the leading coefficient is ad, having the interior factor a ; 
and this being so it is found that all the following coefficients will divide by a (the 
quotients being of course expressible only in terms of the covariants subsequent to 
f): thus the second coefficient of D is — hf-\-ce, and (5.11) we have -bf-\-ce~ai, or 
the coefficient divided by a is —i ; and so for the other coefficients of 1) ; or throwing 
out the factor a, we obtain for D an expression of the form (d, i, . . \x, yf, see the 
Table 98 : this is the “ divided” form of D : and we have similarly a divided form for 
every other capital covariant. All that has been required is that each coefficient of 
the divided form shall be expressed as a rational and integral function of the 
covariants a, b, c ... v, w : and the form is not hereby made definite : to render it 
so the coefficient must be expressed in the segregate form. But there is frequently 
the disadvantage that we thus introduce fractions ; for instance, the last coefficient 
of D is = — ci + df, where to get rid of the congregate term df we have (6.12), 
3 df= — al-\- 2ci, and the segregate form of the coefficient is = — \al-\-fci. 
379. We have in regard to the Canonical form a differential operator which is 
analogous to the two differential operators ocd y — {xd y },yd. r — {yd, r } considered in the 
Introductory Memoir (1854). Let § denote a differentiation in regard to the constants 
under the conditions 
Sa= 0, 
8 b — e, 
Sc = of, 
Sd= - a (-bf+ce), (=i), 
Se = — Gad — lObc, 
8 f= 2adb— 18c 2 , 
which (as at once verified) are consistent with the fundamental relation 
f ' 2 — — a s d-\-a 2 bc — 4c 3 ; 
then it is easy to verify that 
d d 
