320] THE GEOMETRICAL THEORY. 349 



radius of gyration about the remaining conjugate diameter. The data before 

 us are not adequate to the removal of this indefiniteness. It must be 

 remembered that t/he data in such a case are just so many but no more than 

 suffice for the specification of the n-system A. The number of data neces 

 sary to define an n-system is n (6 n). If, as in the present case, n = 4, the 

 number of data is 8. We are thus one short in the number of data necessary 

 to specify a rigid body. Thus we confirm the result previously obtained. 

 We can assert that for any one of the singly infinite number of rigid bodies 

 which fulfil the necessary conditions, the system A will be the locus of the 

 instantaneous screws which correspond to the screws of the system P as 

 impulsive screws. 



Though in the two cylindroids A and P we are able to establish the 

 several pairs of correspondents quite definitely, yet we must not expect, with 

 the data before us, to be able to correlate the pairs of screws in A and P 

 definitely. If this could be done then the rigid body would be quite deter 

 minate, which we know is not the case. There is, however, only a single 

 indeterminate element left in the correlation of the screws in A with the 

 screws of P. This we prove as follows : 



Let &amp;lt; be any screw of P on which an impulsive wrench is to act. Let 8 

 be the instantaneous screw in A about which the movement commences. 

 We shall now show that though 8 cannot be completely defined, in the 

 absence of any further data, yet it can be shown to be limited to a certain 

 cylindroid. 



Let G be the centre of gravity. Then we know that an impulsive force 

 directed through G will generate a movement of translation in a direction 

 parallel to the force. Such a movement may, of course, be regarded as a 

 twist about a screw of infinite pitch. 



Draw through G a plane normal to &amp;lt;/&amp;gt;. Any screws of infinite pitch in 

 this plane will be reciprocal to &amp;lt;. It follows from the laws of conjugate screws 

 of inertia that the impulsive forces in this plane, by which translations could 

 be produced, must lie on screws of zero pitch which are reciprocal to 8. 

 Take any two of such screws : then we know that 8 is reciprocal to these two 

 screws and also to P . It follows that 8 is reciprocal to the screws of a 

 determinate system of the fourth order, and therefore 8 must lie on a deter 

 mined cylindroid. 



We may commence to establish the correspondence between P and A by 

 choosing some arbitrary screw &amp;lt;f&amp;gt; on P, and then drawing the cylindroid on A, 

 on which we know that the instantaneous screw corresponding to P must 

 lie. Any screw on this cylindroid may be selected as the instantaneous 

 screw which corresponds to &amp;lt;/&amp;gt;. Once that screw 8 had been so chosen there 



