360 THE THEORY OF SCREWS. [332, 



332. Co-reciprocal Correspondents in two Three-systems. 



If U be an instantaneous three-system and V the corresponding im 

 pulsive three-system it is in general possible to select one set of three 

 co-reciprocal screws in U whose correspondents in V are also co-reciprocal. 



As a preliminary to the formal demonstration we may note that the 

 number of available constants is just so many as to suggest that some finite 

 number of triads in U ought to fulfil the required condition. 



In the choice of a screw a in U we have, of course, two disposable 

 quantities. In the choice of /3 which while belonging to U is further 

 reciprocal to a there is only one quantity disposable. The screw belonging 

 to U, which is reciprocal both to a and /?, must be unique. It is in fact 

 reciprocal to five independent screws, i.e. to three of the screws of the system 

 reciprocal to U, and to a and /3 in addition. 



We have thus, in general, neither more nor fewer than three disposable 

 elements in the choice of a set of three co-reciprocal screws a, /3, 7 in U. 

 This is just the number of disposables required for the adjustment of the 

 three correspondents rj, , in V to a co-reciprocal system. We might, 

 therefore, expect to have the number of solutions to our problem finite. We 

 are now to show that this number is unity. 



Taking the six principal screws of inertia of the rigid body as the screws 

 of reference, we have as the co-ordinates of any screw in U 



Xa 6 4- fA/3 6 -|- vy 6) 

 where X, p,, v are numerical parameters. 



The co-ordinates of the corresponding screw in V are 



P! (Xctj + fji/Sj. + i/ 7l ), 

 p 2 (Xa 2 + pfa 



where for symmetry p 1} . . . p 6 are written instead of + a, a, + b, b, &c. 

 Three screws in U are specified by the parameters 



X , //, z/; X&quot;, //,&quot;, v&quot;; X &quot;, p&quot;, v&quot;. 

 If these screws are reciprocal, we have 



= 2 P1 (Va, + /& + i/ 7l ) (X&quot; ai + /*&quot;& + *&quot; 7l ), 

 or = \ \&quot;p a + n ^ pt + v v &quot;p y + (X&amp;gt;&quot; + XV) *r*p 



+ (XV + XV) OTay + (ii 

 and two similar equations. 



