58 



THE THEORY OF SCREWS. 



[63- 



If, therefore, AX, BY, OG be perpendiculars on CO , we have, from the 

 principle of the centre of gravity, 



Fig. 14. 



or, p 2 AX + PI BY = ( Pl +PJ OG ; 



but, by a well-known property of the circle, if ra be the radius, 



2mAX = AC . AC -, 2mBY = BC.BC - 



whence 



or 



P1 BC.BC + p.,AC.AG = 2m ( Pi + P ,) OG = m ~ ( 60), 



Ob 



BC_ BW_ AC_ 



pl +P - 



OG 



_ _ _ 



AB AB - AB AB~ m OS 



But, from the expressions for screw co-ordinates ( 57), this reduces to 



* 



The required expression has thus been demonstrated. 



We can give another proof of this theorem as follows : 



If the two screws of reference be reciprocal, and if p l and p 2 be the co 

 ordinates of another screw, then it is known, from the theory of the co 

 ordinates, that the virtual coefficients of this screw, with respect to the screws 

 of reference, are p 1 p l and p.,p, respectively ( 37). 



Thus (Fig. 15) the virtual coefficient of X and A must be ( 57), 



BX 



