340 
A TENTH MEMOIR ON QUANTICS. 
[693 
syzygiës is thus to make each of them give a single congregate in terms of the 
segregates : viz. the left-hand side can always be taken to be a monomial congregate 
a a №... or, to avoid fractions, a numerical multiple of such form; and the right-hand 
side will then be a linear function, with numerical coefficients, of the segregates of 
the same deg-order. Supposing such a system of syzygies obtained for a given deg- 
order, any covariant function (rational and integral function of covariants) is at once 
expressible as a linear function of the segregates of that deg-order : it is, in fact, 
only necessary to substitute therein for every monomial congregate its value as a linear 
function of the segregates. Using the word covariant in its most general sense, the 
conclusion thus is that every covariant can be expressed, and that in one way only, 
as a linear function of segregates, or say in the segregate form. 
Reverting to the theory of the canonical form, and attending to the relation 
f 2 = — a?d + a 2 bc — 4c 3 , 
it thereby appears that every covariant multiplied by a power of the quintic itself a, 
can be expressed, and that in one way only, as a rational and integral function of 
the covariants a, b, c, d, e, f linear as regards f: say every covariant multiplied by 
a power of a can be expressed, and that in one way only, in the “ standard ” form : 
as an illustration, take 
a?h — dacd + 46c 2 -f- ef. 
Conversely, an expression of the standard form, that is, a rational and integral function 
of a, b, c, d, e, f, linear as regards /, not explicitly divisible by a, may very well 
be really divisible by a power of a (the expression of the quotient of course containing 
one or more of the higher covariants g, h, &c.), and we say that in this case the 
expression is divisible, and has for its divided form the quotient expressed as a 
rational and integral function of covariants. Observe that in general the divided form 
is not perfectly definite, only becoming so when expressed in the before-mentioned 
segregate form, and that this further reduction ought to be made. There is occasion, 
however, to consider these divided forms, whether or not thus further reduced; and 
moreover it sometimes happens that the non-segregate form presents itself, or can be 
expressed, with integer numerical coefficients, while the coefficients of the corresponding 
segregate form are fractional. 
The canonical form is peculiarly convenient for obtaining the expressions of the 
several derivatives (Gordan’s Uebereinanderschiebungeri) (a, b)\ (a, b) 2 , &c., (or as I 
propose to write them a61, a62, &c.), which can be formed with two covariants, the 
same or different, as rational and integral functions of the several co variants. It 
will be recollected that in Gordan’s theory these derivatives are used in order to 
establish the system of the 23 covariants : but it seems preferable to have the system 
of covariants, and by means of them to obtain the theory of the derivatives. 
I mention at the end of the Memoir two expressions (one or both of them due 
to Sylvester) for the N.G.F. of a binary sextic. 
The several points above adverted to are considered in the Memoir; the paragraphs 
are numbered consecutively with those of the former Memoirs upon Quantics.