The Decomposition F the General Collineation F Space into Three Skew Reflections

A number of years ago several investigators f published independently this theorem : Any screw motion—the most general mechanical motion of space —can be decomposed into the product of two semi-rotations. By a semi-rotation is meant that transformation which consists of rotating space through 180° about a fixed axis I. For generalizing however it is more convenient to look at this transformation as refection in a line. From this point of view a semirotation may be defined as that transformation which replaces each point P of space by a point P' such that the line PP' cuts orthogonally a fixed line I and is bisected by it. If we turn to the projective group of three dimensions we find in it, as a transformation corresponding to the semi-rotation of the mechanical group, the skew reflection which may be defined as follows : A skew reflection is that transformation of space which replaces each point P by a point P' such that the line PP' cuts each of two non-coplanar lines I, V and is divided harmonically by them. The lines I, I' are called directrices. The purpose of this paper is to ask and answer the question : Is it possible to decompose the general collineation of space into the product of a number oj skew reflections ; and if so, what is the least number of skew reflections involved in such a decomposition ? $ The question will be answered by giving