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 r\ is required, of which 

 the intensity is 



COS 



the mass being for convenience taken as unity. 



Let a, ft, 7 be three of the instantaneous screws in U, and let 77, , be 

 their respective impulsive screws in V. 



Let d, ft, 7 be the component twist velocities on a, ft, 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 77, , f the respective quantities 



7 \ ^&amp;gt; ~ / 7T^\ P) &quot;/ L,\ 7- 



cos (a?7) cos (p) cos (7^) 



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 U as follows. 



Choose any screw other than p in the three- system U and draw the 

 cylindroid H through that screw arid 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 (a/3) drawn through a and ft, 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 7. But by the principle of 292 the several pairs 

 of correspondents on the instantaneous cylindroid (aft) and the impulsive 

 cylindroid (77 ) 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 (77^) will discover the impulsive screw corresponding 

 to another instantaneous screw on H. As therefore we know the impulsive 

 screws corresponding to two of the screws on H we know the cylindroid H 

 which contains all the impulsive screws severally corresponding to instan 

 taneous screws on H, 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. 



