324 THE THEORY OF SCREWS. [303- 



We can now determine the value of p a * where p a is the radius of gyration 

 about an axis parallel to a through the centre of gravity. For the kinetic 

 energy is obviously 



P/d a (/t&amp;gt;a a +^. 2 + a 2 )- 



By equating the two expressions we have 



a 2 = 2 



cos 



But when a and rj are known the three terms on the right-hand side of 

 this equation are determined. Thus we learn the radius of gyration on the 

 diameter parallel to a. 



It remains to show how a certain straight line in the plane which is 

 conjugate to this diameter in the momental ellipsoid is also determined. 

 Let a screw 6, of zero pitch, be placed on that known diameter of the 

 momental ellipsoid which is parallel to or. Draw a cylindroid through the 

 two screws 6 and 77. Let &amp;lt;/&amp;gt; be the other screw of the zero pitch, which will 

 in general be found on the same cylindroid. 



We could replace the original impulsive wrench on 77 by its two com 

 ponent wrenches on any two screws of the cylindroid. We choose for this 

 purpose the two screws of zero pitch 6 and &amp;lt;. Thus we replace the wrench 

 on 77 by two forces, whose joint effect is identical with the effect that would 

 have been produced by the wrench on 77. 



As to the force along the line 6 it is, from the nature of the con 

 struction, directed through the centre of gravity. Such an impulsive force 

 would produce a velocity of translation, but it could have no effect in pro 

 ducing a rotation. The rotatory part of the initial twist velocity must there 

 fore be solely the result of the impulsive force on &amp;lt;. 



But when an impulsive force is applied to a quiescent rigid body we 

 know, from Poinsot s theorem, that the rotatory part of the instantaneous 

 movement must be about an axis parallel to the direction which is conjugate 

 in the momental ellipsoid to the plane which contains both the centre of 

 gravity and the impulsive force. It follows that the ray (f&amp;gt; must be situated 

 in that plane which is conjugate in the momental ellipsoid to the diameter 

 parallel to a. But, as we have already seen, the position of &amp;lt;/&amp;gt; is completely 

 defined on the known cylindroid on which it lies. We have thus obtained a 

 fixed ray in the conjugate plane to a known diameter of the momental 

 ellipsoid. 



The three statements at the beginning of this article have therefore been 

 established. We have, accordingly, ascertained five distinct geometrical data 

 towards the nine which are necessary for the complete specification of the 

 rigid body. These five data are inferred solely from our knowledge of a 

 single pair of corresponding impulsive and instantaneous screws. 



