PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 609 
the terms arising from each such pair are distinct from each other) in the expansion of 
the entire function the coefficients are each = + l. 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 6.11, the expansion 
contains only the terms a~li, accl, bed ; there is besides a term of the same deg-order, 
ef, which is not a term of the expansion, and hence of must be a linear function of 
a~h, acd, be 2 ; we in fact have ef—a/h — Gacd — 46c 3 . 
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_ft 6 ) {l _ w)+ ( 1 _ 6 8 )(o + « )+(l _ 6a) ( c+ * )+(l _ 6 )/ 
+ ( 1 - a( f) (d + h +j + m + dj+hj +/ +jm ) 
+ (1 —bg)(l+j oJ t~js) 
+ (1 — b^g) (i -\-n-\-p -\-j k) 
+ (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 jf into 
(d, A, j, k, m, o, s, t ) where j 3 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 ad/c y (f<pid of 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 
m question, thereby diminishing the number of the segregates : thus as regards the 
terms belonging to unity, any one of these which contains the factor Id 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~ 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 precisely 
equal to the number of congregates : viz., each such syzygy may be regarded as giving 
a congregate in terms of the segregates : we have on the left hand side a congregate, 
4 I 2 
