428 THE THEORY OF SCREWS. [390, 



We also see that 77 must lie on the polar of the point a#i, /3# 2 &amp;gt; 7$s with 

 regard to the same conic. 



We thus obtain a geometrical construction by which we discover the 

 restraining screw when the instantaneous screw is given. 



Two homographic systems are first to be conceived. A point of the first 

 system, of which the co-ordinates are 8 1} # 2 &amp;gt; &s, has as its correspondent a 

 point in the second system, with co-ordinates a0 1} /30 2 , j0 3 . The three 

 double points of the homography correspond, of course, to the permanent 

 screws. 



To find the restraining screw ?? corresponding to a given instantaneous 

 screw 0, we join to its homographic correspondent, and the pole of this 

 ray, with respect to the pitch conic, is the position of 17. 



The pole of the same ray, with regard to the conic of inertia ( 211), is 

 the accelerator. It seems hardly possible to have a more complete geo 

 metrical picture of the relation between 77 and 9 than that which these 

 theorems afford. 



391. Calculation of Permanent Screws in a Three-system. 



When a three-system is given which expresses the freedom of a body 

 we have seen how in the plane representation the knowledge of a conic (the 

 conic of inertia) will give the instantaneous screw corresponding to any 

 given impulsive screw. A conic is however specified completely by five 

 data. The rigid body has nine co-ordinates. It therefore follows that there 

 is a quadruply infinite system of rigid bodies which with respect to a given 

 three-system will have the same conic of inertia. If in that three-system a 

 be the instantaneous screw corresponding to 77 as the impulsive screw for 

 any one body of the quadruply infinite system, then will 77 and a stand in 

 the same relation to each other for every body of the system. 



The point in question may be illustrated by taking the case of a four- 

 system. The screws of such a system are represented by the points in space, 

 and the equation obtained by equating the kinetic energy to zero indicates 

 a quadric. For the specification of the quadric nine data are necessary. 

 This is just the number of co-ordinates required for the specification of 

 a rigid body. If therefore the inertia quadric in the space representation 

 be assumed arbitrarily, then every instantaneous screw corresponding to a 

 given impulsive screw will be determined; in this case there is only a finite 

 number of rigid bodies and not an infinite system for which the correspond 

 ence subsists. 



We thus note that there is a special character about the freedom of the 

 fourth order which we may state more generally as follows. To establish 

 a chiastic homography ( 292) in an n-system requires (n l)(n + 2)/2 data. 



