A Treatise on the Theory of Screws

156 THE THEORY OF SCREWS. [161, We can also determine the chord in a somewhat different manner, which has the advantage of giving certain other expressions that may be of service. Let U = 0 be the cubic curve. Let V= 0 be the equation of the two straight lines from the origin to the points of intersection with the two equal pitch screws + 3. Let L = 0 be the chord joining the two intersections of U and V, distinct from the origin : this is, of course, the chord now sought for. Then we must have an identity of the type cU= VX + LY; where c is some constant. For the conditions L = 0 and K= 0 imply U —0, and L cuts U in three points, two of which lie on V, and the third point, called I, must lie on X. The line X is otherwise arbitrary, and we may, for con- venience, take it to be the line HI from the origin to I. The product YX thus contains only terms of the third degree, and accordingly the terms of the second degree in U must be sought in LY. Let U= u3 + w2 where us and are of the third and second degrees respectively, then cw2 must be the quadratic part of the product LY. As L does not pass through the origin, it must have an absolute term, conse- quently Y must not contain either an absolute term or a term of the first degree. If, therefore, c be the absolute term in L, it is plain that F must be simply w2, and we have accordingly, c (w3 + w2) = VX + (L' + c) u2, where L' denotes the value of L without the absolute term: we have con- sequently the identity cw3 = YX + L’u2. In this equation we know u2, u3, V, and the other quantities have to be found. If we substitute x — — y cos ß tan (a + 6), we make V vanish, and representing L' by Xx + yy, we find „ . sin ß X cos ß tan (a + 3) — y = c 7-7--3-------■—; h tan ß + m sin 26 and after a few steps X __ m cos 2a + m cos 26 c — h2 tan ß + m2 cot ß sin2 26 ’ y h sin ß + m cos ß sin 2a c — h2 tan ß + m2 cot ß sin2 26 ‘