304 THE THEORY OF SCREWS. [284- 



Multiplying by p^, ...p (i a , respectively, and adding, we have 



&amp;lt;*Uaa = ^ &quot;tff^ , 



where u^ = pftf + . . . + p&amp;lt;? &amp;lt;*&amp;lt;? ; 



because, as p belongs to the reciprocal system, we must have 



OT O = 0. 



Similarly, if we multiplied the six equations by p^i, ...p 6 /3 6 , respectively, 

 and added, we should get, since p is reciprocal to /3 also, 



where u aft = p^a^ + . . . + p?a t . 



Eliminating d and rj &quot; we have the concise result*, 



Uaa. &pT, ~ Ua^ar,. 



In a similar way we can deal with the pair of screws, /3 and , and by 

 eliminating &amp;lt;r, the reaction of the constraints in this case, we obtain the 

 result 



Finally, from these two equations we can eliminate w a/3 , and we obtain 



This formula is a perfectly general relation, connecting any two pairs of 

 impulsive and instantaneous screws 77, a and , /3. It holds whether the 

 body be free or constrained in any way whatever. If the body be perfectly 

 free, then it is easy to show that it reduces to the result already found, viz. 



P* PP 



ooBfo) * cos(/3) 



285. Another Proof. 



From the theory of impulsive and instantaneous screws in an n-system 

 we know (97) that if a lf ... ct n be the co-ordinates of an instantaneous screw, 

 then the co-ordinates i) 1} ... 17 w of the reduced impulsive screw may be deter 

 mined as follows : 



tr w i 2 



#,, = -* 



*- v a - 



Pn 



Multiplying severally by p l a l) ... p n a n , and adding, we have 



H^r,a = U aa . 



* Trans. Roy. Irish Acad., Vol. xxx. p. 573 (1894). 



