SCREW CO-ORDINATES. 31 



But taking the screw p in place of 17 we have 



Substituting for sr p(0i &c. TOpWo from the former equa- 

 tions, we deduce 



A p" 



This result may recall the well-known expression for 

 the square of a force acting at a point in terms of its 

 components along three axes passing through the point. 

 This expression is greatly simplified when the three axes 

 are rectangular, and we shall now show that by a special 

 disposition of the screws of reference, a corresponding 

 simplification can be made in the formula just written. 



33- Co-Reciprocal Screws. We have hitherto chosen 

 the six screws of reference quite arbitrarily; we now 

 proceed in a different manner. Take for o>,, any screw ; 

 for w 2 any screw reciprocal to wi ; for a> 3 , any screw 

 reciprocal to ui and w 2 ; for w iy any screw reciprocal to w,, 

 <i> 2 , w 3 ; for w 5 , any screw reciprocal to wi, w 2 , w 3 , w 4 ; for <u 6 , 

 the screw reciprocal to wi, w 2 , w 3 , j 4 , w 5 . 



A group constructed in this way possesses the pro- 

 perty that each pair of screws is reciprocal. Any set of 

 screws not exceeding six, of which each pair is recipro- 

 cal, may be called for brevity a set of co-reciprocals.* 



Thirty constants determine a group of six screws. If 

 the group be co-reciprocal, fifteen conditions must be 

 fulfilled ; we have, therefore, fifteen elements still dis- 

 posable, so that we are always enabled to select a co- 

 reciprocal group with special appropriateness to the 

 problem under consideration. 



* Dr. Klein has discussed (Math. Ann. Band n. p. 204), six linear com- 

 plexes, of which each pair are in involution. If the axes of these complexes be 

 regarded as screws, of which the " auptparameters " are the pitches, then 

 these six screws will be co-reciprocal. 



