A Treatise on the Theory of Screws

2 INTRODUCTION. If we make ;yx = y2 = px.2, y3 = pa;3, y4 = px4 we can eliminate x2, x3, x4 and obtain the following biquadratic for p, (11) -p, (12) (13) (14) (21) (22) -p, (23) (24) (31) (32) (33) -p, (34) (41) (42) (43) (44) — p The four roots of this indicate four double points, i.e. points which remain unaltered. But these points are not necessarily distinct or real. What we have written down is of course the general homographic trans- formation of the points in space. For the displacement of a rigid system is a homographic transformation of all its points, but it is a very special kind of homographic transformation, as will be made apparent when we consider what has befallen the four double points. In the first place, since the distance between every two points before the transformation is the same as their distance after the transformation it follows that every point in the plane at infinity before any finite trans- formation must be in the same plane afterwards. Hence the plane at infinity remains in the same position. Further, a sphere before this transformation is still a sphere after it. But it is well known that all spheres intersect the plane at infinity in the same imaginary circle fl. Hence we see not only that the plane at infinity must remain unaltered by the transformation but that a certain imaginary circle in that plane is also unaltered. A system of points P,, P2, P, &c. on this circle fl will, therefore, have as their correspondent points Qt, Q.2, Q3, &c. also on fl. As all anhar- inonic ratios are unaltered by a linear transformation it follows that the systems Plt P.,, P3, &c. and Qt, Q2, Q:„ &c. are homographic. There will, therefore, be two double points of this homography, 01 and 0.2, and these will be the same after the transformation as they were before. They are, there- fore, two of the four double points of which we were in search. It should be remarked that the points 0, and 0., cannot coincide, for if they coincide in 0, then 0 must be the double point corresponding to a repeated root of the biquadratic for p. But such a root is real. Hence 0 must be real. But every point on £1 is imaginary. Hence this case is impossible. As fl is unaltered and 01 and 02 are fixed, the tangents at 0, and 02 are fixed, and so is therefore T, the intersection of these tangents; this is accordingly the third of the four points wanted. It lies in the plane at infinity, but is a real point. The ray 0j0a is also real; it is the vanishing line of the planes perpendicular to the parallel rays, of which T is the vanishing point.