214 THE THEORY OF SCREWS. [210, 



of course passes through the two points which lie at the intersections of 

 2HR LS = and Q = 0. Hence we deduce the following result. 



The cylindroid through the two screws of zero pitch which intersect a 

 corresponds to the ray whose equation is 



2( 1 2 +a, 2 +a 3 2 )(&amp;gt; 1 a 1 6&amp;gt; 1 +p,aA + p^-A)-(p^ +p,a.* +p&?} (a^ +a 2 2 +a 3 3 )=0, 



a will of course intersect a third screw on this cylindroid and be perpen 

 dicular thereto. 



The cubic passes through the two points defined by 



= aA + oc,&amp;lt;9 2 + a 3 3 , 



but these points are the points of contact of the pair of tangents drawn from 

 the point a to the conic of infinite pitch. 



These two points in addition to the four points on the same conic defined 

 by its intersection with the conic of zero pitch, make six known points on the 

 cubic. If however we are given six points on a cubic and also a double 

 point on the curve, then it is determinate. Thus we obtain the following 

 geometrical construction. 



In the plane representation of a three-system we draw the conic of zero 

 pitch C and the conic of infinite pitch U. To find the locus of the screws 

 which intersect a given screw a, it is necessary to draw the two tangents 

 from a to If and through the two points of contact, and also the four points 

 common to C and U then draw that single cubic which has a double point 

 at a. See Appendix, note 5. 



211. Application to Dynamics. 



By the aid of the plane representation we are enabled to solve certain 

 problems in the dynamics of a rigid body which has freedom of the third 

 order. 



Let an impulsive wrench act upon a quiescent rigid body ; it is required 

 to determine the instantaneous screw about which the body will commence to 

 twist. 



It has been shown ( 96) that the impulsive wrench, wherever situated, 

 can generally be adequately represented by the reduced impulsive wrench 

 on a screw of the three-system. The problem is therefore reduced to the 

 determination of the point corresponding to the instantaneous screw, when 

 that corresponding to the impulsive screw is known. 



