Some properties of line congruences

Abstract

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.