34 SCREW CO-ORDINATES. 



of n sides can be drawn, each of the sides of which is 

 proportional to the amplitude of one of the twists (or in- 

 tensity of one of the wrenches), and parallel to the cor- 

 responding screw. 



Let #, #, , be the direction cosines of aline parallel 

 to any screw of reference w n , and drawn through a 

 point through which pass three rectangular axes. 



Then since a wrench of one unit on a has components 

 of intensities ai, &c. a 6 , we must have 



(#idi + &C. + <2 6 a 6 ) 2 + (b\u } + &C. ^6e) 2 + (^iai & 



whence 



Sa! 2 + 22ai a 2 COS (ai!^) = I, 



if we denote by cos (wi wg) the cosine of the angle between 

 two intersecting lines parallel to on and w 2 . 



38. Calculation of Co-ordinates. We must conceive 

 the formation of a table of triple entry from which the 

 virtual coefficient of any pair of screws may be ascer- 

 tained. The three arguments will be the angle be- 

 tween the two screws, the perpendicular distance, and 

 the sum of the pitches. These arguments having been 

 ascertained by ordinary measurement of lines and 

 angles, the virtual coefficient can be extracted from the 

 tables. 



Let a be a screw, of which the co-ordinates are to be 

 determined. The work done against the unit wrench on 

 a by a twist of amplitude w/ about the screw wi is 



but this must equal the work done by the same twist 

 against a wrench of intensity ai on the screw wj, 

 whence 



-7 



A 



