The projective invariants of four mediale

The theory of four particular linear forms, or matrices of k columns and 2 k rows, occurred to me many years ago in an attempt to study the invariants of any number of compound linear forms, or subspaces within a space of n dimensions. In what follows, the invariant theory is given, and its significance for a study of the general matrix of k rows and columns is suggested. The collineation used in §4 was considered by Mr J. H. Grace, who emphasized the importance of the k cross ratios upon transversal lines of four [ k −1]'s in [2 k −1]. It seemed appropriate to examine these cross ratios which are irrational invariants μ i , of the figure of four such spaces, and to work out their relation to the known rational invariants X i . The main result is given in § 5 (7). In § 5 (10) it is shewn that the harmonic section of a line transversal of the four spaces exists when a linear relation holds between the invariants.