A Treatise on the Theory of Screws

CHAPTER XIII . THE GEOMETRY OF THE CYLINDROID. 159. Another investigation of the Cylindroid. The laws of the composition of twists and wrenches are of such funda- mental importance in the present subject that the following method* of investigating the cylindroid seems worthy of attention. This method is not, however, presented as a substitute for that already given (Chap. II.) which is certainly both more simple and more direct. Let a and ß be any two screws, then if a body receives a twist about a, followed by another twist about ß, the position arrived at could have been reached by a single twist about a third screw 7. If the amplitudes of the twists about a and ß are given, then the position of 7, as well as the ampli- tude of the resultant twist thereon, are, of course, both determined. If, however, the amplitudes of the twists on a and ß are made to vary while the screws a and ß themselves remain fixed, then the position of y, no less than the amplitude of the resultant twist, must both vary. It has however been shown in § 9 that the position and pitch of 7 remain constant so long as the ratio of the amplitudes of the twists about a and ß remains unchanged. As this ratio varies, the position of 7 will vary, so that this position is a function of a single parameter; and, accordingly, 7 must be restricted to be one of the generators of a certain ruled surface 8, which includes a and ß as extreme cases in which the ratio is zero and infinity respectively. Let be a screw which is reciprocal both to a and ß, then 6, must also be reciprocal to every screw 7 on Ä Let 02, 0s, Ot be three other screws also reciprocal to S. Since a screw is defined by five conditions, it is plain that a screw which fulfils the four conditions of being reciprocal to 0lt 02, 03, 04 will have one disposable parameter, and must, therefore, be generally confined to a certain ruled surface. This surface must include 8, inasmuch as all the screws on 8 are reciprocal to 0,, 02, 03, Op, but further, it cannot include any * Proceedings of the Royal Irish Academy, 2nd Ser., Vol. iv. p. 518 (1885).