82] ROLLING POINSOT ELLIPSOID. 259 



diameter is conjugate to Q and lies in the plane conjugate to Q, that 

 is, parallel to the tangent plane at x, y, a. Consequently, if we 

 compound with the velocity co about Q the velocity corresponding to 

 S c dt parallel to the tangent plane, the resultant has the same com- 

 ponent perpendicular to the tangent plane as co. In other words the 

 component to cos (eo, H) is constant throughout the motion. 



Now we have found that H is constant in magnitude and direc- 

 tion, hence, multiplying by the constant ocos(ca#), 



22) Ho cos (G)H) = const. 



But jffcos (Ho) is that component of the angular momentum which 

 is parallel to the instantaneous axis, and is accordingly equal, by 

 68, 53) to the product of the angular velocity by the moment of 

 inertia about the instantaneous axis. 



23) H cos (Hm) = Ko. 

 Accordingly 22) becomes 



24) K a) 2 = const. 



But this is equal to twice the kinetic energy. Accordingly we obtain 

 geometrically the integral of energy. Thus for a rigid body this 

 principle follows from that of the conservation of angular momentum. 

 In the ellipsoid of inertia we have, 71, 



Accordingly 



25) 



and the equation of energy shows that n is constant during the 

 motion, or during the whole motion the angular velocity is propor- 

 tional to the radius vector to the ellipsoid of inertia in the direction 

 of the instantaneous axis. But since cocos(i?(a) is constant, pcos(pd) = d 

 must be constant, and therefore the tangent plane is at a constant 

 distance from the center during the motion. But since the direction 

 of the line d is constant in space, and its length is also constant, 

 the tangent plane must be a fixed plane in space. As the point 

 where it is touched by the ellipsoid of inertia is on the instantaneous 

 axis the ellipsoid must be turning about this radius vector, and hence 

 rolling without sliding on the fixed tangent plane. The motion of 

 the * body is thus completely described, and we see that the problem 

 of a Poinsot- motion is equivalent to the geometrical one of the 

 rolling of an ellipsoid whose center is fixed on a fixed tangent 

 plane, together with the kinematical statement that the angular 

 velocity of rolling is proportional to the radius vector to the point 



17* 



