A Treatise on the Theory of Screws

158 the theory of sckews. [162 The latus rectum is 4 cos ß(h+m cot ß sin 2a). The values of the two equal pitches (p) on the pair of screws are thus expressed in terms of the abscissa a> of the point in which the chord touches the envelope by means of the equation p = Po + tan ß. From any point P, on thß cubic, two tangents can be drawn to thß parabola. Each of these tangents must intersect the cubic in a pair of screws of equal pitch. One tangent will contain the other screw whose pitch is equal to that of P. The second tangent passes through two screws of equal pitch in the two other points which, with P, make up the three intersections with the cubic. As the principal screws of the cylindroid are those of maximum and minimum pitch respectively, it follows that the tangents at these points will also touch the parabola. These common tangents are shown in the figure. This parabola is drawn to scale in Fig. 36. The equation employed was 2/ = _31_(^5+£v) . When the figure was complete, it was obvious that the parabola touched the cubic, and thus the following theorem was suggested:— The parabola, which is the envelope of chords joining screws of equal pitch, touches the cubic in three points. The demonstration is as follows :■—To seek the intersections of the parabola with the cubic, we substitute, in the equation of the parabola, the values x = h tan (0 - a) - m cot ß sin 20 tan (0 - «), y = h sec ß — m cosec ß sin 20. This would, in general, give an equation of the sixth degree for tan 0. It will, however, be found in this case that the expression reduces to a perfect square. The six points in which the parabola meets the cubic must thus coalesce into three, of which two are imaginary. The values of 0 for these three points are given by the equation h tan (0 — a) — m cot ß {sin 20 tan (0 — a) + 2 cos 20} = 0. We can also prove geometrically that the parabola touches the cubic at three points. In general, a cone of screws reciprocal to the cylindroid can be drawn from any external point. If the point 0 happen to lie on the cylindroid,