A Treatise on the Theory of Screws

217] FREEDOM OF THE FOURTH ORDER. 227 If X, p, v be the direction angles of 0 we have thei’efore (§ 44) a (pe — a) cos X — dei sin X C72 =----------------------= 0, a o = 0^ ~ V) cos Z4 ~ sin p = 0 p = (pe-c) cos v - dK sin v = Q c The direction cosines of the common perpendicular to 0 and 1 are cos v cos p ’ sin Å. ’ sin X ’ whence the cosine of the angle between this perpendicular and the radius vector to a point a?, y, z on 0 is y cos v z cos p d61 r ’ sin X r sin X r ’ or dei sin X = y cos v — z cos p. We have thus the three conditions — a) cos X + z cos p — y cos v = 0..........(i), — z cos X + (/>9 — &) cos p + x cos p = 0.........(ii), — y cos X — x cos p 4- (pe — c) cos v = 0.........(iii), whence eliminating cos X, cos p, cos v we obtain (pe - a) (pe - 6) (pe - c) + (pe - a) x2 + (pe - b) t/3 + (pe -c)z2 = 0. Thus we deduce the equation otherwise obtained in § 174, for the family of pitch-hyperboloids on which are arranged according to their pitches the several screws of the three-system. 217. Principal pitches of the Reciprocal Cylindroid. From a system of the fourth order a system of canonical co-reciprocals can in general be selected which possesses exceptional facilities for the investi- gation of the properties of the screws which form that four-system. Let OA and OB be the axes of the two principal screws of the reciprocal cylindroid. Let a and b be the pitches of these two principal screws and let c be any third linear magnitude. Let 00 be the axis of the cylindroid. Then the canonical co-reciprocal system now under consideration consists of Two screws on OA with pitches + a and — a. Two screws on OB with pitches + b and — b. Two screws on 00 with pitches + c and — c. 15—2