A Treatise on the Theory of Screws

364 THE THEORY OF SCREWS. [334- A be a screw in U the impulsive cylindroid, then B will be its instantaneous screw in V the instantaneous cylindroid. It is however to be now shown that there are two screws H1 and 011 U, and their correspondents Æj and K2 on V, which possess the remarkable characteristic that whether V be the impulsive cylindroid and U the instan- taneous cylindroid or vice versa, in either case ll\ and K, are a pair of corre- spondents, and so are II2 and K2. Let B2, B2, B3, &c. be the screws on Ü corresponding severally to the screws Alt A2, A3, &c., on V when V is the impulsive cylindroid and U the instantaneous cylindroid. Let Cj, C2, C3, &c., be the screws on U corresponding severally to the screws A2, A2, A3, &c., on V when U is now the impulsive cylindroid, and V the instantaneous cylindroid. The systems Alt A2, A3, &c., and Blt B2, B3, &c., are homographic. The systems C1; C2, C3, &c., and Alt A2, A3, &c., are homographic. Hence also, The systems B,, B2, B3, &c., and Cj, C2, Cs, &c. are homographic. Let H3, H2 be the two double screws on Ü belonging to this last homo- graphy, then their correspondents Klt K2 on V will be the same whether U be the impulsive cylindroid and V the instantaneous cylindroid or vice versa. There can be no other pairs of screws on the two cylindroids possessing the same property. 335. A Property of Co-reciprocals. Let a, ß, 7 be any three co-reciprocal screws. If £ £ are the three screws on which impulsive wrenches would cause a free rigid body to twist about a, ß, y respectively, then cos (a£) cos (/Si?) cos (?£) + cos (a£) cos ßg) cos (7^) = 0. We have from § 281 the general formula - sfe) °“ <*’+“s but as a and ß are reciprocal each side of this equation must be zero. We thus have —cos (ßß =------------/3p- cos (a£), cos (aß x 7 cos (ßß