A Treatise on the Theory of Screws

60 THE THEORY OF SCREWS. [66- 66. Screws of Zero Pitch. A screw of zero pitch is reciprocal to itself. The tangent at a point corresponding to a screw of zero pitch, being the chord joining two reciprocal screws, must pass through the pole of the axis of pitch. This is, of course, the same thing as to say that the axis of pitch intersects the circle in two screws, each of which has zero pitch. 67. A Special Case. We have supposed that the axis of pitch occupies any arbitrary position. Let us now assume that it is a tangent to the representative circle. This specialization of the general case could be produced by augmenting the pitches of all the screws on the cylindroid, by such a constant as shall make one of the two principal screws have zero pitch. The following properties of the screws on the cylindroid are then obvious:— 1. There is only one screw of zero pitch, 0. 2. The pitches of all the other screws have the same sign. 3. The maximum pitch is double the radius. 4. The screw 0 is reciprocal to every screw on the surface, and this is the only case in which a screw on the cylindroid is reciprocal to every other screw thereon. 68. A Tangential Section of the Cylindroid*. Let the plane of section be the plane of the paper in Fig. 17, and let the plane contain one of the screws of zero pitch OA. Let OH be the projection of the nodal axis on the plane of the paper. Then OA being perpendicular to the nodal axis must be perpendicular to OH. Let P be the point where the second screw of zero pitch cuts the curve. Then since any ray through P and across AO, meets two screws of equal pitch, it must be perpendicular to the third screw which it also meets on the cylindroid (§ 22). Hence PH is perpendicular to the screw through H, and as the latter lies in the norma] plane through OH it follows that the angle at H is a right angle. Any chord perpendicular to zLO must for the same reason intersect two screws of equal pitch, and therefore APHO must be a rectangle. If tangents be drawn at A and P intersecting at T, then it can be shown that any chord TLM through T cuts the ellipse in points L and M on two reciprocal screws. * For proofs of theorems in this article see a paper in the Transactions of the Koval Irish Academy, Vol. xxix. pp. 1—32 (1887).