A Treatise on the Theory of Screws

162] THE GEOMETRY OF THE CYLINDROID. 159 the cone breaks up into two planes. The nature of these planes is easily seen. One of them, A, must be the plane perpendicular to the generator through 0; the other, B, is the plane containing 0, and the screw of equal pitch to that of the screw through 0. These planes intersect in a ray, L, and it must first be shown that L is a tangent to the cylindroid. Any ray intersecting one screw on a cylindroid at right angles must cut the surface again in two screws of equal pitch; consequently L can only meet the surface in two distinct points, each of which has the pitch of the generator through 0. It follows that L must intersect the surface at two coincident points 0, i.e. that it is a tangent to the cylindroid at 0. Let any plane of section be drawn through 0. This plane will, in genera], intersect A and B in two distinct rays: these are the two screws reciprocal to the cylindroid, and they are accordingly the two tangents from 0 to the parabola we have been discussing. The only case in which these two rays could coalesce would occur when the plane of section was drawn through L; but the two tangents to a parabola from a point only coalesce when that point lies on the parabola. At a point where the parabola meets the cubic, L must needs be a tangent both to the parabola and to the cubic, which can only be the case if the two curves are touching. We have thus proved that the parabola must have triple contact with the cubic. There are thus three points on the cubic which have the property that the tangent intersects the curve again in a point of equal pitch to that of the point of contact. We thus learn that all the screws of a four-system which lie in a plane touch a parabola having triple contact with the reciprocal cylindroid. From any point P, on the cubic, two tangents can be drawn to the parabola. Each of these tangents must intersect the cubic in a pair of screws of equal pitch. One tangent will contain the two screws whose pitch is equal to that cf P. The other tangent passes through two screws of equal pitch in the two other points, which, with P, make up the three intersections with the cubic. As the principal screws of the cylindroid are those of maximum and minimum pitch, respectively, it follows that the tangents at these points will also touch the parabola. These common tangents are shown in Fig. 36. From the equation of the cylindroid, z (a? + ?/) = 2mxy, it follows that the plane at infinity cuts the surface in three straight lines on the planes, z = Q, x + iy = 0.