358 
A TENTH MEMOIR ON QUANTICS. 
[693 
and we have thus the expression of В (see the Table No. 97) ; and similarly for the 
other capital covariants G, D, .., V, W : in every case the coefficients are obtained 
in the standard form, that is, as rational and integral functions of a, b, c, d, e, f, 
linear as regards f 
378. It will be observed that there is in each case a certain power of a which 
explicitly divides all the coefficients and is consequently written as an exterior factor: 
disregarding these exterior factors, the leading coefficients for B, G, D, E, F are 
h, c, ad, e, f respectively; that for G is 12abd + 46 2 c + e 2 , which must be = g multi 
plied by a power of a, and (in Table 97) is given as = a?g ; similarly, that for H is 
Qacd + 4be 2 + ef, which must be = h multiplied by a power of a, and is given as = аЯь : 
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, G, E, F there is no power 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, G, 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 /): thus the second coefficient of D is —bf+ce, and (5.11) 
we have — bf+ce = ai, or the coefficient divided by a is = i; and so for the other 
coefficients of D; or throwing out the factor a, we obtain for D an expression of 
the form (d, i,...\x, y) 3 , 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 i) is =- ci + df y where to get rid of 
the congregate term df we have (6.12), 3df= — al + 2ci, and the segregate form of 
the coefficient is = — |al + § ci. 
379. We have in regard to the canonical form, a differential operator which is 
analogous to the two differential operators xd y — {xd y }, yd x — [yd^ considered in the 
Introductory Memoir (1854), [139]. Let 8 denote a differentiation in regard to the 
constants under the conditions 
8a = 0, 
8b = e, 
8c = 3f 
U= i(-bf+ce), (= i), 
CL 
8e = — 6ad — 10 be, 
8f = 2a% - 18c 2 ,