308 THE THEORY OF SCREWS. [290- 



about some corresponding screw on A, and the two systems of screws would 

 have chiastic homography. If the body were given both in constitution 

 and in position, then, of course, there would be nothing arbitrary in the 

 choice of the corresponding screws. Suppose, however, that a screw i) had 

 been chosen arbitrarily on B to correspond to a screw a on A, it would 

 then be generally possible to design and place a rigid body so that it should 

 begin to twist about a in consequence of the impulse on ??. There would, 

 however, be no arbitrary element remaining in the homography. Thus, we 

 see that, while for homography, in general, three pairs of correspondents 

 can be arbitrarily assigned, there can only be two pairs so assigned for 

 chiastic homography, while for such a particular type as that which relates 

 to impulsive screws and the corresponding instantaneous screws, only one 

 pair can be arbitrarily chosen. 



291. Case of Normal Cylindroids. 



We have already had occasion ( 118), to remark on the curious relation 

 ship of two cylindroids when a screw can be found on either cylindroid which 

 is reciprocal to all the screws on the other. If, for the moment, we speak 

 of two such cylindroids as &quot; normal,&quot; then we have the following theorem : 



Any homography of the screws on two cylindroids must be chiastic if 

 the two cylindroids are normal. 



Let a, /3, 7 be any three screws on one cylindroid, and 77, , any three 

 screws on the other ; then, since the cylindroids are normal, we have 



whence we obtain 



GJ &amp;gt;f ( /nr af 5J 0i) 5 J &quot;yf sf at af P 1s yr,) = , 



unless therefore is reciprocal to /3, we must have 



SJ a^^ SJ-yf ^a^ft^yyt = 0. 



If, however, had been reciprocal to /3, then one of these screws (suppose /3) 

 must have been the screw on its cylindroid reciprocal to the entire group of 

 screws on the other cylindroid. In this case we must have 



5% = ; ts p$ = 0, 

 so that even in this case it would still remain true that 



. = 0. 



It is, indeed, a noteworthy circumstance that, for any and every three pairs 

 of screws on two normal cylindroids, the relation just written must be 

 fulfilled. 



