A Treatise on the Theory of Screws

234 THE THEORY OF SCREWS. [224, 224. Properties of a Quadratic Two-system. Ihe quadratic two-system is constituted of screws whose coordinates satisfy three linear equations and one quadratic equation, and these screws lie generally on a surface of the sixth degree (§ 225). If we take the plane representation of the three-system given in Chapter XV., then any conic in the plane corresponds to a quadratic two-system and all the points in the plane correspond to the enclosing three-system. Since any straight line in the plane corresponds to a cylindroid in the enclosing system and the straight line will, in general, cut a conic in the plane in two points, we have the following theorem. A quadratic two-system has two screws in common with any cylindroid belonging to the enclosing three-system. A pencil of four rays in the plane will correspond to four cylindroids with a common screw, which we may term a pencil of cylindroids. Any fifth transversal cylindroid belonging also to the same three-system will be intersected by a pencil of four cylindroids in four screws, which have the same anharmonic ratio whatever be the cylindroid of the three-system which is regarded as the transversal. We thus infer from the well-known anharmonic property of conics the following theorem relative to the screws of a quadratic two-system. If four screws a, ß, y, c be taken on a quadratic two-system, and also any fifth screw y belonging to the same system, then the pencil of cylindroids (i/a), (yß), (yy), (yb) will have the same anharmonic ratio whatever be the screw y. (See Appendix, note 6.) The plane illustration will also suggest the instructive theory of Polar screws which will presently be stated more generally. Let U= 0 be the conic representing the quadratic two-system and let F= 0 be the conic representing the screws of zero pitch belonging to the enclosing three-system. Let P be a point in the plane corresponding to an arbitrary screw 0 of the three-system. Draw the polar of P with respect to U= 0 and let Q be the pole of this straight line with respect to V= 0, then Q will correspond to some screw </> of the enclosing three-system. From any given screw 0, then by the help of the quadratic two-system a corresponding screw </> is determined. We may term </> the polar screw of 0 with respect to U= 0. Three screws of the enclosing system will coincide with their polars. These will be the vertices of the triangle which is self-conjugate with respect both to U and to K A possible difficulty may be here anticipated. The equation V= 0 is itself of course equivalent to a certain quadratic two-system and therefore should correspond to a surface of the sixth degree. We know however (§ 173) that the locus of the screws of zero pitch in a three-system is au hyperboloid, so that in this case the expectation that the surface would rise to the sixth