A Treatise on the Theory of Screws

28 THE THEORY OF SCREWS. [23, screws on the cylindroid intersected by the perpendicular, then the perpen- diculars form a cone of reciprocal screws. We shall now prove that this cone is of the second order, and we shall show how it can be constructed. Let 0 be the point from which the cone is to be drawn, and through 0 let a line OT be drawn which is parallel to the nodal line, and, therefore, perpen- dicular to all the generators. This line will cut the cylindroid in one real point T (Fig. 4), the two other points of intersection coalescing into the in- finitely distant point in which OT intersects the nodal line. Draw a plane through T and through the screw LM which, lying on the cylindroid, has the same pitch as the screw through T. This plane can cut the cylindroid in a conic section only, for the line LM and the conic will then make up the curve of the third degree, in which the plane must intersect the surface. Also since the entire cylindroid (or at least its curved portion) is included between two parallel planes (§17), it follows that this conic must be an ellipse. We shall now prove that this ellipse is the locus of the feet of the per- pendiculars let fall from 0 on the generators of the cylindroid. Draw in the plane of the ellipse any line TUV through T- then, since this line intersects two screws of equal pitch in T and U, it must be perpendicular to that generator of the cylindroid which it meets at V. This generator is, therefore, perpendicular to the plane of OT and VT, and, therefore, to the line OV. It follows that V must be the foot of the perpendicular from 0 on the