44] SCREW CO-ORDINATES. 39 



we have 



2aa l = ( a+p a )\&quot;- (p&quot;z - v&quot;y ), 



similarly 2aa 2 = (- a + p a ) A.&quot; (fjif z v&quot;y ), 



we thus find 



\&quot; = a, + a., ; v&quot;y pf z = a (i a,) p a (a, -f a,). 



In like manner we obtain two similar pairs of equations for the required 

 equations of the screw a, 



(a.-, + ?) y - (a, f 04) z = a (j - ou) - p a (^ + o 2 ), 



(a, -f a,) z - ( 6 4- ) a; = 6 (a, - a 4 ) - &amp;gt; (a s + 4 ), 



(i). 



(a s + 4 ) # - (i + ,) y=c(a s - a 6 ) - ^ a ( 5 + a 6 



The expressions on the right-hand side of these equations are the co 

 ordinates of the extremity of a vector from the origin of length equal to 

 the perpendicular distance of a from the centre, and normal to the plane 

 containing both a and the origin. 



The co-ordinates of the foot of the perpendicular from the origin on the 

 screw a are easily shown to be 



x - ( 5 - 6 ) ( + a 4 ) c - (a 5 + a,) ( 3 - a 4 ) b, 

 i/ = (di- a,) ( 3 + O a - (! 4- a ) ( 5 - ) c, 

 ^ = (a s - a,) (a, -1- a,) b - (a 3 + 4 ) (otj - a 2 ) a, 



44. A Screw of Infinite Pitch. 



The conception of the screw co-ordinates as defined in 41 require special 

 consideration in the case of a screw of infinite pitch. Consider a wrench on 

 such a screw. If the intensity of the wrench be one unit, then the 

 moment of the couple which forms part of the wrench is infinite. As the 

 pitches of the screws of reference or any of those pitches are not in general 

 to be infinite, it follows that the wrench of unit intensity on a screw of 

 infinite pitch must have for its components on one or more of the screws of 

 reference wrenches of infinite intensity. 



If therefore a u 2 , ... a 6 be the co-ordinates of a wrench of infinite pitch, 

 it is essential that one or more of the quantities ci l} a.,, ... a s shall be 

 infinite. 



In the case where the screws of reference form a canonical system we can 

 obtain the co-ordinates as follows : 



(p a + a) cos (al) - d Al sin (al) _ (/&amp;gt; a - a) cos (al) - rf al sin (rl) 

 ~~ ; &quot; 2 ~ - 



