A Treatise on the Theory of Screws

178] FREEDOM OF THE THIRD ORDER. 179 This equation denotes a form of Steiner’s surface : where a — 2x b — c + -2,J- + c — a 2z a — b + 1, 2x b — c c — a 2z a — b + 1, 7 = 2x b — c + _ty_ _ c — a 2z a — b + 1, 3 = 2x b — c c — a _2z_ a — b + 1. From the form of its equation it appears that this surface has three double lines which meet in a point, viz. the three axes OX, OY, OZ. This being so any plane will cut the surface in a quartic curve with three double points, being those in which the plane cuts the axes. If the plane touch the surface, the point of contact is an additional double point on the section, that is, the section will be a quartic curve with four double points, i.e. a pair of conics. The projections of the origin on the generators of any cylindroid belonging to the system lie on a plane ellipse (§ 23). This ellipse must lie on the Steiner quartic. Hence the plane of the ellipse must cut the quartic in two conics and must be a tangent plane. See note on p, 182. 178. Screws of the Three-System parallel to a Plane. Up to the present we have been analysing the screw system by classifying the screws into groups of constant pitch. Some interesting features will be presented by adopting a new method of classification. We shall now divide the general system into groups of screws which are parallel to the same plane. We shall first prove that each of these groups is in general a cylindroid. For suppose a screw of infinite pitch normal to the plane, then all the screws of the group parallel to the plane are reciprocal to this screw of infinite pitch. But they are also reciprocal to any three screws of the original reciprocal system; they, therefore, form a screw system of the second order (§ 72)—that is, they constitute a cylindroid. We shall prove this in another manner. A quadric containing a line must touch every plane passing through the line. The number of screws of the system which can lie in a given plane is, therefore, equal to the number of the quadrics of the system which can be drawn to touch that plane.