136 DYNAMICS OF A RIGID BODY. 



the same time reciprocal to TJ, must all lie upon a cylin- 

 droid, as they each fulfil the condition of being recipro- 

 cal to four screws. All the screws on the cylindroid 

 are parallel to a certain plane drawn through the centre 

 of the pitch quadric, which may be termed the reciprocal 

 plane with respect to the screw TJ. The reciprocal plane 

 having been found, the diameter conjugate to this plane 

 in the ellipsoid of inertia is parallel to the required 

 screw 6. 



For let JJL and v denote two screws of the complex 

 parallel to a pair of conjugate diameters of the ellipsoid 

 of inertia in the reciprocal plane. Then 0, p, v are 

 a triad of conjugate screws of inertia ; but r? is reciprocal 

 to fj. and v, and, therefore, by the lemma of the last 

 article, an impulsive wrench upon r\ will make the body 

 commence to twist about 0. 



123. Kinetic acqEnergy uired by an Impulse. We 

 shall now consider the following problem : A quies- 

 cent rigid body of mass M receives an impulsive wrench 

 of intensity rj" on a screw TJ for a short time e. De- 

 termine the locus of a screw 8 belonging to a screw 

 complex of the third order, such that, if the body be con- 

 strained to twist about 0, it shall acquire a given kinetic 

 energy K, in consequence of the impulsive wrench. 



We have from 6 1 the equation 

 



& ri" 2 



K = ^r -V 

 M uf 



We can assign a geometrical interpretation to this 

 equation, which will lead to some interesting results. 



Through the centre O of the pitch quadric the plane 

 A reciprocal to t\ is to be drawn. A sphere ( 1 1 6) is 

 to be described touching the plane A at the origin O, 

 the diameter of the sphere being so chosen that the 

 intercept OP made by the sphere on a radius vector 



