302, 303] THE GEOMETRICAL THEORY. 323 



position of rest by an impulsive system of forces of the most general type. 

 This is the object of the present chapter. 



303. One Fair of Impulsive and Instantaneous Screws. 



Let it be supposed that nothing is known of the position, mass, or other 

 circumstances of an unconstrained rigid body save what can be deduced 

 from the fact that, when struck from a position of rest by an impulsive 

 wrench on a specified screw 77, the effect is to make the body commence to 

 move by twisting around a specified screw a. 



As a, like every other screw, is defined by five coordinates, the knowledge 

 of this screw gives us five data towards the nine data that are required for 

 the complete specification of the rigid body and its position. 



We have first to prove that the five elements which can be thence 

 inferred with respect to the rigid body are in general 



(1) A diameter of the momental ellipsoid. 



This is clearly equivalent to two elements, inasmuch as it restricts 

 the position of the centre of gravity to a determinate straight line. 



(2) The radius of gyration about this diameter. 



This is, of course, one element. 



(3) A straight line in the plane conjugate to that diameter. 



A point in the plane would have been one element, but a straight 

 line in the plane is equivalent to two. If the centre of gravity were 

 also known, we should at once be able to draw the conjugate plane. 



Draw a plane through both the instantaneous screw a and the common 

 perpendicular to a and 77. Then the centre of gravity of the rigid body 

 must lie in that plane ( 301). It was also shown that if p a be the pitch of 

 a, and if (ar)) represent the angle between a and 77, then the perpendicular 

 distance of the centre of gravity from a. will be expressed by p a tan ((277) 

 ( 301). This expression is completely known since a and 77 are known. 

 Thus we find that the centre of gravity must lie in a determinate ray 

 parallel to a. There will be no ambiguity as to the side on which this 

 straight line lies if it be observed that it must pass between a and the point 

 in which 77 is met by the common perpendicular to 77 and a. In this manner 

 from knowing a and 77 we discover a diameter of the momental ellipsoid. 



If a be the twist velocity with which a rigid body of mass M is twisting 

 about any screw a. If 77 be the corresponding impulsive screw, and if Ts ar&amp;gt; 

 denote as usual the virtual coefficient of a and 77, then it is proved in 279 

 that the kinetic energy of the body 



MCt 2 -, r GJ an- 



cos (077) 



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