352 THE THEORY OF SCREWS. [323 



follows, that a wrench on the screw of zero pitch, which lies on the ray 

 G l G 2 , will have the same instantaneous screw whether that wrench be 

 applied to M or to M . 



We have now to examine whether there can be any other pair of im 

 pulsive and instantaneous screws in the same circumstances. Let us suppose 

 that when 6 assumes a certain position 77, we have A, and /u, coalescing into the 

 single screw a. 



We know that the centre of gravity lies in a plane through a, and the 

 shortest distance between a and 77. We know, also, that d a =p a tan (a??), 

 where d a is the distance of the centre of gravity from a. It therefore follows 

 that a must be parallel to GiG 2 . We have, however, already had occasion 

 ( 303) to prove that, if p a be the radius of gyration of a body about a ray 

 through its centre of gravity, parallel to a, 



2p a sr a ,, , 



pa 2 = -^f \ da ~ Pa- 



cos (377) 



Hence it appears that, for the required condition to be satisfied, each of the 

 two bodies must have the same radius of gyration about the axis through its 

 centre of gravity, which is parallel to a. Of course this will not generally be 

 the case. It follows that, in general, there cannot be any such pair of 

 impulsive screws and instantaneous screws, as has been supposed. Hence we 

 have the following result : 



Two rigid bodies, with different centres of gravity, have, in general, no other 

 common pair of impulsive screws and instantaneous screws than the screw, of 

 zero pitch, on the ray joining the centres of gravity, and the screw of infinite 

 pitch parallel thereto. 



We shall now consider what happens when the exceptional condition, just 

 referred to, is fulfilled, that is, when the radius of gyration of the ray G 1 G 2 is 

 the same for each of the bodies. 



In each of the momental ellipsoids about the centres of gravity of the 

 two bodies, draw the plane conjugate to the ray G^G 2 . Let these planes 

 intersect in a ray T. Suppose that an impulsive force, directed along T, be 

 made to act on the body whose centre of gravity is O l . It is plain, from 

 Poinsot s well-known theory, that the rotation produced by such an impulse 

 will be about a ray parallel to G 1 G 2 . If this impulsive wrench had been 

 applied to the body whose centre of gravity is G 2 , the instantaneous screw 

 would also be parallel to G 1 G 2 . If we now introduce the condition that the 

 radius of gyration of each of the bodies, about G 1 G 2 , is the same, it can be 

 easily deduced that the two instantaneous screws are identical. Hence we 

 see that T, regarded as an impulsive screw of zero pitch, will have the same 

 instantaneous screw for each of the two bodies. 



