A Treatise on the Theory of Screws

212 THE THEORY OF SCREWS. [209, Ihe following are the pitches of the several points and curves represented Vertices of the Triangle +'992, +'825, -448, Ellipses + -96, + 92, + 1, - A, Parabola + ’309, Hyperbolas + '748984, 4- -8300467. The locus of the centres of the pitch conics is aAsin (A - B) sin 2(7 + a3a3 sin (B - C) sin 2A + a.^ sin (<7- A) sin 2B = 0. 210. Intersecting Screws in a Three-System. If two screws of a three-system intersect, then their two corresponding points must fulfil some special condition which we propose to investigate. Let a be one screw supposed fixed, then we shall investigate the locus of the point 0 which expresses a screw which intersects a. We can at once foresee a certain character of this locus. A ray through a can only cut it in one othei point, for if it cut in two points, we should have three co-cylindroidal screws intersecting, which is not generally possible. The locus is, as we shall find, a cubic and the necessary condition is secured by the fact that a is a double point on the cubic, so that a ray through a has only one more point of intersection with the curve. We can indeed prove that this curve must a cubic fiom the fact that any serßw niGcts a cylmdroid in throe points. Draw then a ray (§ 200) corresponding to a cylindroid of the three-system. There must be in general three points of the locus on this ray. Therefore the locus must be a cubic. As a and 0 intersect we have since d6a = 0 2-^9« = (pa +Pe) cos (a0). By substituting the values of the different quantities in terms of the co- ordinates we have the following homogeneous equation of the cubic: 0 = 2 (Ö,2 + 022 + 02) (p^Ø, +p2a202 +p3a303) (a? + a22 + as2) - (PiOi +p2022 +p3032) (*Ä + a2é»2 + a303) (a? + a22 + a32) - (022 + 02 + 0S2) + j>2a22 +/>3as2) (ajØj + a.202 + a303). We first note that this cubic must pass through the four points of inter- section of O = 02+02+032, o = pA2 + p2fy? 4-ps032. But this might have been expected, because as we have shown (§ 203) each of these four points corresponds, not to a single screw, but to a plane of screws.