A Treatise on the Theory of Screws

162] THE GEOMETRY OF THE CYLINDROID. 157 We can also obtain X; for let it be Px + Qy, then severally identifying the coefficients of Xs and y'\ we have, P = m sin 2a + h tan ß, Q = — m cos ß (cos 2a + cos 20); finally, resuming the various results, we obtain the identity cU^VX + LY-, wherein, c = — h2 tan ß + m2 cot ß sin2 20, U = sin ß cos2 ßy2 + sin ßyx2, — (m sin 2a + h tan ß) x2 + 2mxy cos ß cos 2a — (m sin 2a cos2 ß — h sin ß cos ß) y2. V = mx2 (cos 2a 4- cos 20) + 2mxy sin 2a cos ß + my2 cos2 ß (cos 20 — cos 2a), X = x (m sin 2a + h tan ß) — my cos ß (cos 2a + cos 20), L = mx (cos 2a + cos 20) + y (h sin ß + m cos ß sin 2a) — h2 tan ß + m2 cot ß sin- 20, Y = x2 (— m sin 2a — h tan ß) + 2mxy cos ß cos 2a + y2 (m sin 2a cos2 ß — h sin ß cos ß). 162, Parabola. The screws reciprocal to a cylindroid intersect two screws of equal pitch on the surface. Any chord in the section which cuts the cubic in two points of equal pitch must thus be the residence of a screw reciprocal to the surface; accordingly the chord mx (cos 2a + cos 20) + y (h sin ß + m cos ß sin 2a) — h2 tan ß + m2 cot ß sin2 20 = 0, when it receives a pitch equal to — p„ — in cos 20, forms a screw reciprocal to the cylindroid. It is easily shown that the envelope of this chord is a parabola; differ- entiating with respect to 0 we have x = 2m cot ß cos 20. Eliminating 0 we obtain x2 + 4>mx cot ß cos 2a + 4sy (h cos ß + m cos ß cot ß sin 2a) — 4A2 + 4nt2 cot2 ß = 0. The vertex of the parabola is at the point x = — 2m cot ß cos 2a ; y = h sec ß — m cosec ß sin 2a.