The cogredient and digredient theories of multiple binary forms

Abstract

The theory of invariants originally confined itself to forms involving a single set of homogeneous variables; but recent investigations, geometric as well as algebraic, have proved the importance of the study of forms in any number of sets of variables. In passing from the theory of the simple to the theory of the multiple forms, an entirely new feature presents itself: in the latter case the linear transformations which are fundamental in the definition of invariants may be the same for all the variables or they may be distinct, i. e., the sets of variables involved may be cogredient or digredient. Multiple forms thus have two distinct invariant theories, a cogredient and a digredient. The object of this paper is to study the relations between these two theories in the case of forms involving any number of binary variables. Geometrically, such a form may be regarded as establishing a correspondence between the elements of two or more linear manifolds; in the digredient theory the latter are considered as distinct, thus undergoing independent projective transformations, while in the cogredient theory the linear manifolds are considered to be superposed, thus undergoing the same projective transformation. The first part of the paper, ?? 1-5, is devoted to the double forms. The extension of the results is made first, for convenience of presentation, to the triple forms in ? 6, and then to the general case in ? 7. The case of the double binary forms is perhaps the most interesting geometrically. In addition to the general interpretation by means of an algebraic correspondence between two manifolds, such a form may be interpreted as an algebraic curve on a quadric surface, or as a plane algebraic curve from the view point of inversion geometry. In the former of these special interpretations the two binary variables are the (homogeneous) parameters of the two sets of generators on the quadric, while in the latter they are the parameters of the two sets of minimal lines in the plane. These interpretations suggest the