A Treatise on the Theory of Screws

INTRODUCTION. 3 In general in any homographic transformation there cannot be four distinct double points in a plane, unless every point of the plane is a double point. 01 suppose Pj, P2, P3, Pt were four distinct coplanar double points and that any other point R had a correspondent R'. Draw the conic through Plt P2, 1 R. 1 hen R must lie on this conic because the anharmonic ratios i. P2, Ps, Pt) and R'(Plt P2t P3> P4) are equal. We have also Pi(P2, P3, Pit R) and P1(P.2, P3t P4> R') equal, but this is impossible if R and R be distinct. R is therefore a double point. In the case of the displaced rigid body suppose there is a fourth distinct double point in the plane at infinity. Each ray connected with the body will then have one double point at infinity, so that after the transformation the ray must again pass through the same point, i.e. the transformed position of each ray must be parallel to its original position. This is a special form of displacement. It is merely a translation of the whole rigid system in which every ray moves parallel to itself. In the more general type of displacement there can therefore be no double point distinct from T, (\, 0, and lying in the plane at infinity. Nor can there be in general another double point at a finite position T'. For if so, then the ray T1 is unaltered in position, and any finite point T" on the lay TT’ will be also unaltered, since this homographic transformation does not alter distances. Hence every point on TT’ is a double point. Here again we must have fallen on a special case where the double points instead of being only four have become infinitely numerous. In this case every point on a particular ray has become a double point. The change of the body from one position to the other could therefore be effected by simple rotation around this ray. There must however be four double points even in the most genera] case. Not one of these is to be finite, and in the plane at infinity not more than three are to be distinct. The fourth double point must be in the plane at infinity, and there it must coincide with either Olt 02 or T. Thus we learn that the most general displacement of a rigid system is a homographic trans- formation of all its points with the condition that two of its double points are on the imaginary circle fl in the plane at infinity, while the pole of their chord gives a third. Of these three one, we shall presently see which one, is to be regarded as formed of two coincident double points. All rays through T are parallel rays, and hence we learn that in the general displacement of a rigid body there is one real parallel system of rays each of which L is transformed into a parallel ray L'. Let A be any plane perpendicular to this parallel system. Let L and L' cut A in the points R and R'. Then as L and L' move, R and R' are corresponding points in two plane homographic systems. Any two such systems in a plane will of course 1 — 2