336 THE THEORY OF SCREWS. [310, 



Two conjugate points to conic (i) denote two reciprocal screws. Two con 

 jugate points to conic (ii) denote two screws at right angles. 



Let A be any point representing an instantaneous screw. Take the 

 polar of A, with respect to conic (i). Let P be the pole of this ray, with 

 respect to conic (ii). 



Then P will correspond to the impulsive screw, while A corresponds to 

 the appropriate instantaneous screw. For this is clearly a homography of 

 which A and P are two correspondents. Further, the double points of this 

 homography are the vertices of the common conjugate triangle to conies (i) 

 and (ii). If A lie on (i), then its polar is the tangent to (i) ; and as every 

 point on this polar will be conjugate to P, with respect to conic (ii), it 

 follows that A and P are conjugate, with respect to (ii) that is, A and P 

 are correspondents of a pair of screws at right angles. As the pitch of the 

 screw, corresponding to A, is zero, we have thus obtained the solution of 

 our problem. 



311. Fundamental Problem with Free Body. 



We now give the geometrical solution of the problem so fundamental 

 in this present theory which may be thus stated : 



A perfectly free body at rest is struck by an impulsive wrench upon a 

 given screw. It is required to construct the instantaneous screw about which 

 the body will commence to twist. 



The rigid body being given, its three principal axes are to be drawn 

 through its centre of gravity. The radii of gyration a, b, c about these 

 axes are to be found. On the first principal axis two screws of pitches + a 

 and a respectively are to be placed. Similarly screws of pitches + b, b, 

 and +c, c are to be placed on the other two principal axes. These are, of 

 course, the six principal screws of inertia: call them A 0&amp;gt; A l} A. 2 , A 3 , A 4&amp;gt; A 5 . 

 We then draw the five cylindroids 



AoA 1} A A 2 , A A 3&amp;gt; A Q At, A A B . 



It is always possible to find one screw on a cylindroid reciprocal to any 

 given screw. In certain cases, however, of a special nature, more than a 

 single screw can be so found. Under such circumstances the present 

 process is inapplicable, but the exceptional instances will be dealt with 

 presently. 



Choose on the cylindroid A A l a screw 1 which is reciprocal to the 

 given impulsive screw ?;, which is, of course, supposed to lie anywhere and 

 be of any pitch. 



In like manner, choose on the other four cylindroids screws 2 , 3 , # 4 , 6 6 , 

 respectively, all of which are also reciprocal to 77. 



