298] DEVELOPMENTS OF THE DYNAMICAL THEORY. 317 



Let X be the screw on A which is reciprocal to P, 

 S .................. P ........................ A. 



Then any screw //, on A and any screw cj&amp;gt; on P fulfil the conditions 



WA$ = 0, TOV,, = 0. 



Hence &amp;lt; is the impulsive screw corresponding to fj, as the instantaneous 

 screw. 



298. Three Pairs of Correspondents. 



Let a, T\\ ft, J; ; 7, be three pairs of impulsive and instantaneous screws ; 

 let 9, &amp;lt;/&amp;gt; be another pair. Then, if we denote by L a p = 0, and M a p = 0, the 

 two fundamental equations 



C08 



Pa Pfi 



cos (XT ** ~ cos 



we shall obtain six equations of the type 



L 6a = 0, L 9li = 0, L^ = 0, 

 Jlfe a = 0, Jlftf = 0, M ey = 0. 



From these six it might be thought that &amp;lt;f&amp;gt; lt ... &amp;lt; 6 could be eliminated, 

 and thus it would, at first sight, seem that there must be an equation for 6 

 to satisfy. It is, however, obvious that there can be no such condition, for 6 

 can of course be chosen arbitrarily. The fact is, that these equations have 

 a peculiar character which precludes the ordinary algebraical inference. 



Since ,;; ft, ; 7, ; are three pairs of screws, fulfilling the necessary 

 six conditions, a rigid body can be adjusted to them so that they are 

 respectively impulsive and instantaneous. We take the six principal screws 

 of inertia of this body as the screws of reference. We thus have, where 

 p a , pp, p y are certain factors, 



By putting the co-ordinates in this form, we imply that they satisfy the 

 six equations of condition above written. 



Substituting the co-ordinates in L 0a = 0, we get 

 = + (a, + a 2 ) (pefa + p 9 fa) + (a, + a,) (p e &amp;lt;f&amp;gt; 3 + p e $ t ) + ( + a.) (p e &amp;lt;j&amp;gt; t 

 + (0i + 0a) (ai - 2&amp;gt; + (0 + 04&amp;gt; (b&amp;lt;*&amp;gt; ~ 6*4) + (0 5 + 6 ) (c 8 - ca 6 ) 

 - 2 (aa^ - a^6 2 ) - 2 (6o0 - ba,e 4 ) - 2 (c 5 ^ 5 - ca,0). 



