362 THE THEORY OF SCREWS. [333, 



will have its corresponding impulsive screw defined by the equations 



( 326) 



+ ear]! = pi 2 + a? ; - ea^ = p x 2 - a 2 ; 



+ ebrjs = az bz - I? ; ebrj 4 = az + bz l s 2 ; 

 + ecrj 5 = cy - ay -1 2 2 ; - ecr) 6 = - ay - cy - Z 2 2 . 

 By substituting these in the equations just given, we obtain 



az - bz - l s 2 _ . a? + pi* ^ a 2 - pi 2 

 b a a 



- az n - bz + 1 3 2 , a? + p? , a 2 - Pl 2 



- - . - 

 a 



cy - ay - I? = &amp;gt; , a 2 + pi 2 



ay + cy, + 1? _ A&amp;gt; ,, a? + p, 2 + B// , d? - pf 

 c a a 



In like manner from the screw on U 



0, 0, 1, 0, 0, 

 we obtain 



+ arj l = az Q bz ti I/ , ur)-&amp;gt; = az bz / 3 2 ; 



+ brj 3 = p, 2 + 6 2 ; -6174= p-?-b*; 



+ crj 5 = bx CX l*\ Crf = bx + CC If. 



Introducing these into the equations for 77, we have 



az - bz - I/ 7 az + bz + 1 3 2 

 = A -- --- h x&amp;gt; 



a a 



6 2 - p 2 2 . , az - bz - 1 3 2 D , az + bz 

 , r = A +n 



a a 



- bz - Z 3 2 az + bz 



r 



caa 

 - bx - cx + /j 2 . , az - bz - l&amp;lt;? n ,,, az + bz + 1 3 * 



J\. i -*- 



c a a 



Thus we have eight equations while there are nine co-ordinates of the 

 rigid body. This ambiguity was, however, to be expected because, as proved 

 in 306, there is a singly infinite number of rigid bodies which stand to the 

 two cylindroids in the desired relation. 



The equations, however, contain one short of the total number of co 

 ordinates ; aa , y , z , I, 2 , I, 2 , 1 3 2 , p, 2 , p 2 2 are all present but p 3 2 is absent. 



