III.—The Invariant Theory of the Correlation

Abstract

Beyond the ternary and quaternary cases considered by Clebsch (1891) and Weitzenböck (1910), there appears to be no writing on the bilinear form in cogredient variables—the correlation—from the point of view of the projective invariant theory. In the present work a complete system of concomitants is found for the single bilinear ground form. Incidentally it is shown that such a system belongs not to the single form but to the simultaneous system of two mutually conjugate forms. Symbolic methods are used, and the processes are analogous to those employed in the cases of two quadratic forms (Turnbull and Williamson, 1929), and also of one bilinear form (Turnbull, 1932) in contragredient variables. In contrast with these previous results it is remarkable that the present system involves no forms of weight greater than two (§ 3, § 6).