A TENTH MEMOIR ON QUANTICS. 
345 
where the first factor is the entire series of terms sc s q*ut, and the second factor is 
the series of terms a a №gy omitting only those terms which are divisible by abg: and 
in the product of the two factors the terms are all distinct, so that the coefficients 
are still each = 1. The same thing is true for every other pair of numerator terms: 
and since the terms arising from each such pair are distinct from each other, in 
the expansion of the entire function the coefficients are each = + 1. Hence (as in 
the case of the quartic) for any given deg-order, the terms in the expansion of the 
R.G.F. may be taken for the asyzygetic covariants of that deg-order; and if there 
are any other terms of the same deg-order, each of these must be a linear function, 
with numerical coefficients, of these asyzygetic covariants: thus deg-order 6.14, the 
expansion contains only the terms a% acd, be 2 ; there is besides a term of the same 
deg-order, ef, which is not a term of the expansion, and hence ef must be a linear 
function of a~h, acd, be 2 ; we in fact have ef = a 2 h — 6acd — 4>bc 2 . 
The terms in the expansion of the R.G.F. may be called “ segregates,” and the 
terms not in the expansion “ congregates ”; the theorem thus is: every congregate is 
a linear function, with determinate numerical coefficients, of the segregates of the same 
deg-order. 
369. I stop to remark that the numerator of the R.G.F. may be written in the 
more compendious form 
(l_^)(l_ v ) + (l_63)( 0 + i) + (l-& 2 )( e +^) + (l-5)/ 
+ (1 — ag 2 ) (d + h + j + in + dj + hj +j 2 +jm) 
+ (! ~ h 9) ( l +jo+js) 
+ (1 — b 2 g) (i + n+p +jk) 
4- (1 — abg) s 
+ (i -g)jt 
+ (1 - a) w ; 
but the first-mentioned form is, I think, the more convenient one. 
370. It is to be noticed that the positive terms of the numerator are unity, the 
seventeen covariants d, e, f h, i, j, k, l, m, n, o, p, r, s, t, v, w, and the products of j by 
(d, h, j, k, m, o, s, t), where j 2 is reckoned as a product; in all, 26 terms. Disregarding 
the negative terms of the numerator the expansion would consist of these 26 terms, 
each multiplied by every combination whatever a a №cyg s q*uS 0 f the denominator terms 
a, b, c, g, q, u (which for this reason might be called “ reiterative ”): the effect of the 
negative terms of the numerator is to remove from the expansion certain of the terms 
in question, thereby diminishing the number of the segregates: thus as regards the 
terms belonging to unity, any one of these which contains the factor b 5 is not a 
segregate but a congregate : and so as regards the terms belonging to d, any one of 
these which contains the factor ag 2 is a congregate: and the like in other cases. 
For a given deg-order we have a certain number of segregates and a certain 
number of congregates: and the number of independent syzygies of that deg-order is 
C. X. 44