1878.] 



Prof. A. Cayley on Quanlics. 



453 



side will then be a linear function, with numerical coefficients, of the 

 segregates of the same deg-order. Supposing such a system of syz'y- 

 gies 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 neces- 

 sary 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 general 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 / 2 = — a 3 d + a?bc — 4<c 3 , it thereby appears that every co- 

 variant 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, /, linear as regards /; 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, we may take a 2 h= 

 6acd + 4&c 2 + ef. Conversely, an expression of the standard form, that 

 is, a rational and integral function of a, b, c, d, e, /, 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 becom- 

 ing so when expressed in the before-mentioned segregate form, and 

 that this further reduction ought to be made. There is occasion, how- 

 ever, to consider these divided forms, whether or not thus further 

 reduced, and moreover it sometimes happens that the form presents 

 itself or can be obtained with integer numerical coefficients, while the 

 coefficients of the corresponding segregate form are fractional. 



The canonical form is peculiarly convenient for obtaining the ex- 

 pressions of the several derivatives (Gordan's " Uebereinanderschie- 

 bungen"), (a, b) 1 , (a, b) 2 , &c. (or, as I propose to write them, abl, 

 ab2, &c), which can be formed with two covariants the same or 

 different, as rational and integral functions of the several covariants. 

 It will be recollected that, in Grordan'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.Gr.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. 



