A Treatise on the Theory of Screws

180 THE THEORY OF SCREWS. Q178, The quadric surface whose equation is (P* — k)xä + (pß - k) y2 + {py - k) z2 + (pa - k) (pß - Æ) (py - k) = 0, touches the plane Px + Qy + Rz + 8 = 0, when the following condition is satisfied: ?3(Pß - {Py - *) + Q^Pa - k) (py - k) + R\pa-k)(pt-k) + S2 = 0; whence it follows that two values of k can be found, or that two quadrics can be made to touch the plane, and that, therefore, two screws of the system, and, of course, two reciprocal screws, lie in the plane. From this it follows that all the screws of the system parallel to a plane must in general lie upon a cylindroid. For, take any two screws parallel to the plane, and draw a cylindroid through these screws. Now, this cylindroid will be cut by any plane parallel to the given plane in two screws, which must belong to the system; but this plane cannot contain any other screws; therefore, all the screws parallel to a given plane must lie upon the same cylindroid. 179. Determination of a Cylindroid. We now propose to solve the following problem:—Given a plane, deter- mine the cylindroid which contains all the screws, selected from a screw system of the third order, which are parallel to that plane. Draw through 0 the centre of the pitch quadric a plane A parallel to the given plane. We shall first show that the centre of the cylindroid required lies in A (§ 174). Let Tlt T2 (Fig. 37) be two points in which the two quadrics of constant pitch touch the plane of the paper, which may be regarded as any plane parallel to A ; then P is the intersection of the pair of screws belonging to the system PTlt P'l\, which lie in that plane, and P' is the intersection of the pair of reciprocal screws P’RS, P'R., belonging to the reciprocal