320 THE THEORY OF SCREWS. [299- 



necessity for this arrangement is thus shown. If not constant, then there 

 would generally have been some screw , for which tsr a ^ was zero. In this 

 case, of course, ^ yn would be generally zero also. But 7 and tj being both 

 given, this is of course not generally true. The only escape is for r af 

 to be constant. 



300. A difficulty removed. 



Given a and 77, @ and and also 7, then the plane of is determined 

 from the equations of the last article. 



As OT a and ts^ are constant, both a and /3 must be parallel to the plane 

 already considered. But as an impulsive screw could not be reciprocal to an 

 instantaneous screw, it would seem that w y ^ must never be zero, but this 

 condition can only be fulfilled by requiring that must be parallel to 

 the same plane. Whence a, /3, 7 must be parallel to the same plane. But 

 these three screws are quite arbitrary. Here then would seem to be a 

 contradiction. 



The difficulty can be explained as follows : 



Each rigid body, which conformed to the condition that a, /3 and 77, f 

 shall be two pairs of corresponding impulsive and instantaneous screws, will 

 have a different screw corresponding to a given screw 7. Thus, among the 

 various screws , in the degraded cylindroid, each will correspond to one 

 rigid body. In general, of course, it would be impossible for f to be 

 reciprocal to 7. It would be impossible for an impulsive wrench to make a 

 body twist about a screw reciprocal thereto. Nevertheless, it seemed certain 

 that, in general, there must be a screw reciprocal to 7. For otherwise, 

 a, /3, 7 should be all parallel to a plane, which, of course, is not generally 

 true. If, however, a, or b, or o were zero, then the body will have no 

 mass ; consequently no impulse would be necessary to set it in motion. 

 This clearly is the case when is reciprocal to 7. We have thus got over 

 the difficulty. and 7 are reciprocal, in the case when the rigid body is 

 such that a, or 6, or c is zero. 



301. Two Geometrical Theorems. 



The perpendicular from the centre of gravity on any instantaneous screw 

 is parallel to the shortest distance between that instantaneous screw and the 

 corresponding impulsive screw. 



The perpendicular from the centre of gravity on any instantaneous screw 

 is equal to the product of the pitch of that screw, and the tangent of the angle 

 between it and the corresponding impulsive screw. 



