A Treatise on the Theory of Screws

167] THE GEOMETRY OF THE CYLINDROID. 167 The principal screws on the cylindroid both pass through the double point; the two tangents to the curve at this point must therefore both touch the parabola. The leading feature of the central section is expressed by the important property possessed by the chords joining reciprocal screws. If we add any constant to the pitches, then we alter X, and, accordingly, the point P, through which all reciprocal chords pass, moves along the curve. The tangent to the cubic at P meets the cubic again in the point reciprocal to P. Two tangents, real or imaginary, can be drawn from P to the cubic touching it in the points 1\, Ta, respectively: as these must each correspond to a screw reciprocal to itself, it follows that 1\ and T, are the screws of zero pitch. We hence see that the two tangents from any point on the cubic touch the cubic in points of equal pitch. Let a and ß be two screws, and 7 and 8 another pair of screws, and let the two chords, aß and 78, intersect again on the cubic. If d and 0 be the perpendicular distance and angle between the first pair, and d' and 0’ the corresponding quantities for the second pair, then there must be some quantity a>, which, if added to all the pitches on the cylindroid, will make « and ß reciprocal, and also 7 and 8 reciprocal. We thus have (p„ + Pß + 2 w) cos 0 — d sin 0 = 0, (py + Ps + 2<w) cos 0' — d' sin 0' = 0 ; whence, pa +Pß- d tan 0 =py+ps — d' tan 0'; in other words, for every pair of screws, a and ß. whose chords belong to a pencil diverging from a common point on the surface, the expression pa + Pß — d tan 0 is a constant. The value of this constant is double the pitch of the screw of either of the points of contact of the two tangents from P to the curve. 167. Section Parallel to the Nodal Line. If the node on the cubic be at infinity, the form of equation to the cubic hitherto employed will be illusory. The nature of this section must therefore be studied in another way, as follows:— Let the plane cut the two perpendicular screws in A and B. Let I be the perpendicular OC from 0 upon C, and let 7/ be the inclination of this perpendicular to the axis of x. Then, taking Ozl as the new axis of x, in which case z will be the new y, we have x = I tan (y — 0), y' = m sin 20.