316 THE THEORY OF SCREWS. [296- 



from it by d^. In like manner if &amp;lt;/&amp;gt; be an impulsive screw corresponding to 

 yu, as instantaneous screw we have another locus parallel to p for the centre 

 of gravity. 



But as the centre of gravity is at infinity these two loci must there 

 intersect, i.e. they must be parallel and so must X and fi, and hence all 

 instantaneous screws must be parallel. 



Thus we see that all the screws on A must be parallel, i.e. that A must 

 have degraded into an extreme type of cylindroid. 



297. Another extreme Case. 



.Given any two cylindroids A and P it is, as we have seen, generally 

 possible to correlate in one way the several screws on A to those on P so 

 that an impulsive wrench given to a certain rigid body about any screw on 

 P would make that body commence to move by twisting about its cor 

 respondent on A. One case of failure has just been discussed. The case 

 now to be considered is not indeed one of failure but one in which any 

 two pairs of screws on A and P will stand to each other in the desired 

 relations. 



Suppose that A and P happened to fulfil the single condition that each 

 of them shall contain one screw which is reciprocal to the other cylindroid. 

 We have called the cylindroids so circumstanced &quot; normal.&quot; 



Let A, be the screw on A which is reciprocal to every screw on P. If 

 then P and A are to stand to each other in the required relation, A. must 

 be reciprocal to its impulsive screw. But this is only possible on one 

 condition. The mass of the body must be zero. In that case, if there is 

 no mass involved any one of the screws on P may be the impulsive screw 

 corresponding to any one of the screws on A . 



Here again the question arises as to what becomes of the homographic 

 equation which defines so precisely the screw on P which corresponds to 

 the screws on A ( 295). It might have been expected that in the case 

 of two normal cylindroids this homographic equation should become evan 

 escent. But it does not do so. 



But there is no real contradiction. The greater includes the less. 

 If every screw on P will suit as correspondent every screw on A then d 

 fortiori will the pairs indicated by the homography fulfil the conditions 

 requisite. 



That any two pairs of screws will be correspondents in this case is 

 obvious from the following. 



