47] SCREW CO-ORDINATES. 41 



The direction of the screw of infinite pitch is indicated by the fact that 

 as a twist about it is a translation with components a (i - 2 ), b (a 3 a 4 ), 

 c ( g 6 ), the screw must be parallel to a ray of which these three quantities 

 are proportional to the direction cosines. 



As the three equations to the screw have disappeared, the situation of 

 the screw is indeterminate. This is of course what might be expected, 

 because a couple is equally efficacious in any position in its plane. 



46. A Screw at infinity. 



If we have 



&amp;lt;*! + 2 = ; 3 + a 4 = ; 5 + = : 



then the three equations (i), of (43) will be satisfied for a screw entirely at 

 infinity, no matter what its pitch may be. From this and the last article we 

 see that the three equations 



j + , = ; a 3 + a 4 = ; a, + % = 



may mean either a screw of infinite pitch and indefinite position, or a screw 

 of indefinite pitch lying in the plane at infinity. 



47. Screws on one axis. 



The co-ordinates being referred to six canonical co-reciprocals, it is required 

 to determine the co-ordinates of the screws of various pitches which lie on the 

 same axis as a given screw a. 



We have from 36 



_ (a + j?.)(i + . 2 ) - &amp;lt; sin (al) 



~ 



_ (i + a ) - d ai sin (ai) 



~ 



~~ 

 whence ^ w 1 = (^ + tt,). 



2Q 



We thus have the useful results 



6)5 = a& ~ 



These formulae may be verified by observing that one of the equations ( 43) 

 defining o&amp;gt; is 



(&&amp;gt; 5 + a&amp;gt; 9 ) y (w 3 + w 4 ) s = a (a^ o&amp;gt; 2 ) p m (coj -f o) a ). 



