Bicombinants of the rational plane quartic and combinant curves of the rational plane quintic

Abstract

Each covariant locus of a rational plane curve contributes its share toward the covariant and invariant tlleorST of rational curves of higher order by reason of the fact that osculantst of rational curves play the same role in regard to rational curves as polars do in regard to binary forms. Especially is this true of covariant points and covariant lines of the rational curve of order n (whicll we shall call Rn) because corresponding to each covariant line or point of all Rn is a series of covariant rational curves of any Rk, where D > n; the parametric equations of these covariant rational curves are easy to obtain. In tlle first section of this paper a new kind of covariant loclls is defined for Rn and illustrated by important loci associated witll tlle R4. Section II is devoted to combinant curves of the R5. The theory of combinant curves of the R5 in the plane corresponds to the invariant theory of the R6 in space; tllis correspondence is pointed out and illustrated. Finally a sufficient number of combinants and combillative sl-iCS of two line sections of tlle plane R) have been foulld to form the algebraic basis of an analytic treatment of combinant curves of the Ro in the plane, which is shown to be identical with the illvariant theory of the Ro in space.