A Treatise on the Theory of Screws

331] VARIOUS EXERCISES. 359 We know (§ 279) that to generate the unit twist velocity on an instan- taneous screw a an impulsive wrench on the screw y is required, of which the intensity is cos (a,;) ’ the mass being for convenience taken as unity. Let a, ß, 7 be three of the instantaneous screws in U, and'let y, %, £ be their respective impulsive screws in V. Let ä, ß, y be the component twist velocities on a, ß, y of a twist velocity p on any other screw p, belonging to the system U. Then the impulsive wrench on V, which has p as its instantaneous screw will have as its components on £ the respective quantities P* Pß ö Py cos (ay) ’ cos (ߣ) ’ cos (y£) Z' These are accordingly the co-ordinates of the required impulsive wrench. 331. Geometrical Solution of the same Problem. When three pairs of correspondents in the two impulsive and instan- taneous systems of the third order V and U are known we can, in general, obtain the impulsive screw in V corresponding to any instantaneous screw p in Ü as follows. Choose any screw other than p in the three-system II and draw the cylindroid II through that screw and p. Every screw on a cylindroid thus obtained must of course belong to U. Then H must have a screw in common with the cylindroid (aß) drawn through a and ß, for this is necessarily true of any two’ cylindroids which lie in the same three-system. In like manner H must also have a screw in common with the cylindroid (ay) drawn through a and y. But by the principle of § 292 the several pairs of correspondents on the instantaneous cylindroid (aß) and the impulsive cylindroid (y%) are determined. Hence the impulsive screw corresponding to one of the screws in H is known. In like manner the known pairs on the two cylindroids (ay) and (yl?) will discover the impulsive screw corresponding to another instantaneous screw on II. As therefore we know the impulsive screws corresponding to two oi the screws on H we know the cylindroid II which contains all the impulsive screws severally corresponding to instan- taneous screws on II, of which of course p is one. But by § 293 we can now correlate the pairs on H and H', and thus the required correspondent to p is obtained.