Ball — On Problems in the Dynamics of a Rigid Body. 433 



conic is the same straight line as the polar of 6 loith regard, to the inertia 

 conic. 



If ^be tlie virtual coefficient of two screws 6 and t/, then 



It follows that the locus of the points which have a given virtual co- 

 efficient with a given point is a conic ^ touching the conic of infinite 

 pitch at two points. If i/^ be the screw whose polar with regard to 

 the infinite-pitch conic is identical with the polar of 17 with regard to 

 the zero-pitch conic, then all the screws 6 which make a given virtual 

 coefficient with 7/ are equally inclined to i/^. It hence follows that all 

 the screws of a three-system which have a given virtual coefficient 

 with a given screw are parallel to the generators of a right circular 

 cone. All the screws reciprocal to tj form a cylindroid, and i/^ is the 

 one screw of the system which is parallel to the nodal line of the 

 cylindroid. The virtual coefficient of xj/ and 7] is greater than that of 

 T] with any other screw. 



If ^ be a screw about which, when the body is twisting with a 

 given twist velocity it has a given kinetic energy, then we must have 



Mi^^i'- + u^-e^~ + u^e^ - 7r(^i' + 0,' + ^3^) .= 0, 



where h^ is a constant proportional to the energy. It follows that the 

 locus of 6 must be a conic constantly passing through the four points 

 of intersection of 



Oi- + e,^ + Oi' - 0. 



The four points in which these two conies intersect correspond to the 

 screws about which the body can twist with indefinite kinetic energy. 

 These four points A, B, C, D being known, the kinetic energy appro- 

 priate to every point P can be readily ascertained. It is only neces- 

 sary to measure the anharmonic ratio subtended by P, ai A, B, C, D, 

 and to set off distances u^, u-?, u^, h^ on a straight line, so that the an- 

 harmonic ratio of the four points shall be equal to that subtended by 

 P. This will determine 7r, which is proportional to the kinetic energy 

 of the unit twist velocity about the screw corresponding to P. 



A quiescent rigid body of mass Jf receives an impulsive wrench of 

 given intensity on a given screw 77 ; determine the locus of the screw 6 

 belonging to the three-system, such that if the body be constraiaed to 

 twist about ^, it shall acquire a given kinetic energy. 



2 Tlie non-Euclidian geometer will regard all such conies as " circles." 



