A Treatise on the Theory of Screws

182 THE THEORY OF SCREWS. [179- drawn through T' parallel to the external and internal bisectors of the angle between the asymptotes are the two rectangular screws of the cylindroid. Thus the problem of finding the cylindroid is completely solved. It is easily seen that each cylindroid touches each of the quadrics in two points. We may also note that a screw of the system perpendicular to the plane passes through T. Thus given any cylindroid of the system the position of the screw of the system parallel to the axis of the cylindroid is determined*. 180. Miscellaneous Remarks. We are now in a position to determine the actual situation of a screw 0 belonging to a screw system of the third order of which the direction is given. The construction is as follows:—Draw through 0 the centre of the pitch quadric a radius vector OR parallel to the given direction of 0, and cutting the pitch quadric in R. Draw a tangent plane to the pitch quadric in R. Then the plane A through OR, of which the intersection with the tangent plane is perpendicular to OR, is the plane which contains 0. For the section in which A cuts the pitch quadric has for a tangent at R a line perpendicular to OR; hence the line OR is a principal axis of the section, and hence (§179) one of the two screws of the system in the plane A must be parallel to OR. It remains to find the actual situation of 0 in the plane A. Since the direction of 0 is known, its pitch is determinate, because it is inversely proportional to the square of OR. Hence the quadric can be constructed, which is the locus of all the screws which have the same pitch as 0. This quadric must be intersected by the plane A in two parallel * In a letter (10 April 1899) Professor C. Joly writes as follows:—Any plane through the origin contains one pair of screws A and B belonging to the system intersecting at right angles and another pair A’ and B' belonging to the reciprocal system. The group A, B, A', B' form a rectangle of which the origin 0 is the centre. The feet of the perpendiculars from 0 on A and on B and the point of intersection of A' and B' will lie on the Steiner’s quartic (b - c)2y2z2 + (e - a)2z2x2+(a - b)2x2y2= +(b - c) (c-a) (a-b)xyz. The point of intersection of A and B and the feet of the perpendiculars on A' and B' will lie on the new Steiner’s quartic (ft - c)2 y2z2 + (e - a)2 z2x2 + (a - 6)2 x2y2= - (b - c) (c - a) (a - b) xyz. The locus of the feet of the perpendiculars on the screws of a three-system from any arbitrary origin whatever is still a Steiner’s quartic, but its three double lines are no longer mutually rect- angular. They are coincident with the three screws of the reciprocal three-system which passed through the origin. This quartic is likewise the locus of the intersection of the pairs of screws of the reciprocal system which are coplanar with the origin. There is a second Steiner’s quartic whose double lines coincide with the three screws of the given system which pass through the origin and which is the locus of intersection of those pairs of screws of the given system which lie in planes through the origin. It is also the locus of the feet of perpendiculars on the screws of the reciprocal system.