Coventiarr.—Instead of a function of the coefficients only, we may have a function of the coefficients and variables, possessed of the like property of remaining unaltered to a factor prh by the linear transfer motion : such function is termed a covariant. Thus, a covariant (tho Hessian) of the quartic (a, b, c, d, c) (x, y)' is (a c, a e + 2 d-3 b c —2c d, c The quantic itself is one of its own covariants. The term covariant may be used in contradistinction to, or as including, invariant. Tho terms Invariant, covariant, have been explained in reference to the simple case of a single quantio containing but one set of variable'', but they apply equally to the case of a system of quantics, and to quanties which are homogeneous functions of two or more distinct sets of variables. There is one case which it is proper to mention : if in con* junction with a quantic (*) (x, y, z . .)? we consider a linear function x +n y+ Cz+. , the invariants of the system are functions of the coefficients of the quantic, and of the coefficients t,s,C . . . of the linear function ; and treating these as facients, the invariant is said to be a contravarient of the given quantic.
The foregoing definition gives the characteristic property of a variant, but it does not directly show how the covariants of a given quantic are to be investigated. This is supplied as follows :—For any quantic with arbitrary coefficients, for example (a, b, c,d) (x, y)r, there exist operators involving differentiations in respect to the coefficients, tantamount to the operators x el, and y d, in respect to the variables ; thus the operator a db +2 b d., + 3 edd is tantamount to y d,, and 3 b d, + 2 b d, + c dd is tantamount to xd,. Or what is the same thing, denoting for shortness these operators by t y d, cly 1 respectively, the quantic is reduced to zero by each of the operators d, —yd,, lx Any function of the coefficients and t e variables which, in like manner with the quantic itself, is reduced to zero by these two operators respectively, is said to be a covariant ; or, if it contains the coefficients only, an invariant of the quantic.
That the two definitions lead to the same result is of course a theorem to be proved.
The leading coefficient of a covariant, say the coefficient of x. in any covariant of a binary quantic (*) (x, y)', possesses the property of being reduced to zero by the operator t y d, ), and has been termed a peninrariant, but a more appropriate term is seminvariant. An in variant is a function of a given degree in the coefficients,and a covariant is a function of a given degree iu the coefficients and order in the variables, and they may be and are designated accordingly ; thus, the above-mentioned invariants I, J of a binary quartic are called respect ively the quadrinrariant and the cubinvariant, and the covariant of the same quartic is termed the quadricovariant, or if the distinction were required it would be termed the quadricovariant quartic. In these cases the designations are sufficient, but it is to be noticed that in general there is more than one invariant or covariant of the same degree or of the same degree and order, and that any such designation is only a generic, not a specific, name. An invariant or covariant may also be designated by a name referring to the mode of generation—for example, the discriminant. The name catalecticant denotes a certain invariant of a binary quantic of an even order : namely, it is a determinant, which, for the above-mentioned quartic function, is— I a, b, c b, c, d c, d, e (being in this particular case the cubinvariant), and the name canoni sant denotes a certain covariant of a binary quantic of an odd order, namely, it is a determinant the terms whereof are linear functions of the coefficients, and which for the cubic (a ,b, c, d) (x, y)s is lax +by,bx+ cg I lbx + cy, cx + dy (being for the particular case the Hessian or quadricovariant).