DYNAMICS OF A RIGID BODY. 137 



parallel to any screw is equal to ^9 ( 1 16). The quan- 

 tity u is inversely proportional to the radius vector OQ 

 of the ellipsoid of inertia, which is parallel to ( 119). 

 Hence for all the screws of the screw complex which 

 acquire a given kinetic energy in consequence of a 

 given impulse, we must have the product OP. OQ con- 

 stant. 



From a well-known property of the sphere, it follows 

 that all the points Q must lie upon a plane A', parallel 

 to A. This plane cuts the ellipsoid of inertia in an 

 ellipse, and all the screws required must be parallel 

 to the generators of the cone of the second degree, 

 formed by joining the points of this ellipse to the 

 origin, O. 



Since we have already shown how, when the direc- 

 tion of a screw belonging to a screw complex of the 

 third order is given, the actual situation of that screw is 

 determined ( 1 1 5), we are now enabled to ascertain all 

 the screws on which the body acted upon by a given 

 impulse would acquire a given kinetic energy. 



The distance between the planes A and A' is pro- 

 portional to OP. OQ, and therefore to the square root of 

 K. Hence, when the impulse is given, the kinetic energy 

 acquired on a screw determined by this construction is 

 greatest when A and A' are as remote as possible. For 

 this to happen, it is obvious that A' will just touch 

 the ellipsoid of inertia. The group of screws will, there- 

 fore, degenerate to the single screw parallel to the dia- 

 meter of the ellipsoid of inertia conjugate to A. But we 

 have seen ( 122) that the screw so determined is the 

 screw which the body will naturally select if permitted 

 to make a choice from all the screws of the complex of 

 the third order. We thus see again what Euler's theorem 

 ( 64) would have also told us, viz., that when a quies- 

 cent rigid body which has freedom of the third order is 



