A Treatise on the Theory of Screws

CHAPTER II. THE CYLINDROID. 9- Introduction. Let a and ß be any two screws which we shall suppose to be fixed both ln Position and in pitch. Let a body receive a twist of amplitude a' about a’ flowed by a twist of amplitude ß' about ß. The position attained could have been arrived at by a single twist about some third screw p with an amplitude p. We are always to remember that the amplitudes of the twists are infinitely small quantities. With this assumption the order in which the twists about a and ß are imparted will be immaterial in so far as the resulting displacements are concerned. The position attained is the same whether a follows ß' or ß' follows a'. Any change in a' or in ß' will of course generally entail a change both in the pitch and in the position of p. It might thus seem that p depended upon two parameters, and that consequently the different positions of p would form a doubly infinite series, known in linear geometry as a congruence. But this is not the case, for we prove that p depends only upon the ratio of a' to ß' and is thus only singly infinite. lake any point P and let ha be the perpendicular distance from P to a, while pa is as usual the pitch of the screw; then the point P is transferred by the twist about a through the distance Vp? + V a'. The twist about ß conveys P to a distance + h? ß'. The resultant of these two displacements conveys P in a direction which depends upon the ratio of a to ß', and not upon their absolute magnitudes. Let P and Q be two points on p, then the resultant displacement will convey P and Q to points P' and Q' respectively which are also on the axis