A Treatise on the Theory of Screws

________ _________________________________________________________________ I g the theory of screws. ly> of p Suppose that a and ß' be varied while their ratio is preserved P and Q will then be transferred to P" and Q" while by the property just proved P P' P" will be collinear and so will Q, Q', Q"- It therefore follows that as P, P', Q, Q’ are collinear so will P, Q, P", Q." be collinear. The line PQ will therefore be displaced upon itself for every pair of values a and ß which retain the same ratio. The position of the resultant screw is thus not altered by any changes of a! and ß', which preserves their ratio. Let w be the angle between a and ß. We take the case of a point P at an infinite distance on the common perpendicular to a and ß. 1 ns point is displaced through a distance equal to h Va'2+/37?+ 2a'/3' cos &>, where h stands for the infinite perpendicular distance from P to a or to ß. This displacement of P is normal to p which itself intersects at right angles the common perpendicular to a and ß. As the perpendicular distance from P to p can only differ by a finite quantity from h bp' = A Va'2 + ß'2 + 2a ß' cos &>, ___________ _____ p' = Va'2 -I- ß'2 + 2a'ß’ cos w. This determines the amplitude of the resulting twist which is, it may be noted, independent of the pitches. Let d> be the angle between the directions in which a point Q on p is displaced by the twists about a and ß, then the square of the displacement of Q will be _________ (p2 + V) a'2 + (pf + V) ß'2 + 2 V/V + V vaß' cos </>; but this may also be written pp2 (a'2 + ß'2 + 2a ß' cos cP), whence we see that pp depends only on the ratio of a to ß. The pitch and the position of p thus depend on the single numerical parameter expressing the ratio of a' and ß'. As this parameter vanes so will p vary, and it must in successive positions coincide with the several generators of a certain ruled surface. Two of these generators will be the situations of a and of ß corresponding to the extreme values of zero and infinity respectively, which in the progress of its variation the parameter will assume. We shall next ascertain the laws according to which twists (and wrenches) must be compounded together, that is to say, we shall determine the single screw, one twist (or wrench) about which will produce the same effect on t e _______________________