604 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
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 l 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 cdh=6acd-\- 4:bc 2 -\-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 f 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 co variants 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 co variants. 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 Uebereinanderschiebungen ) (a, b) 1 , (a, b ) 3 , &c. (or as I 
propose to write them ab 1, ab2, &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 co variants : 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 Qualities. 
