336 Rotation of a Solid Body round a Fixed Point. 



The first two of these equations prove that the axis of rota- 

 tion is the perpendicular on tangent plane of the ellipsoid, and 

 the last equation gives the magnitude of the rotation in terms 

 of the impressed couple and quantities determined by the nature 

 of the body itself. Equations (9) are true, whatever be the 

 forces acting on the body ; if no forces act, G will be fixed in 

 magnitude and position in space, by the principle of conserva- 

 tion of areas, but will change its position in the body, the axis 

 of rotation accompanying it, and changing its position both in 

 the body and in space. 



VI. ROTATION PRODUCED BY CENTRIFUGAL FORCE ; PARTICU- 

 LAR PROPERTIES OF THE MOTION WHEN NO FORCES ACT. 



The axis of rotation produced by the centrifugal couple always 

 lies in the plane of principal moments. This theorem may be 

 thus proved : Let the radius vector and perpendicular be drawn, 

 which coincide with the axis of principal moment and axis of ro- 

 tation at any instant ; a line perpendicular to the plane of radius 

 vector and perpendicular is the axis of centrifugal couple ; 

 this line and the original radius vector are axes of the section of 

 the ellipsoid made by their plane : at the point where the axis 

 of the centrifugal couple pierces the ellipsoid let a tangent plane 

 be applied ; the perpendicular let fall on this tangent plane is 

 the axis of rotation produced by centrifugal forces. From the 

 construction it is evident that the plane of the second radius 

 vector and perpendicular is perpendicular to the axis of G ; hence 

 the axis of the centrifugal couple and the axis of rotation pro- 

 duced by it always lie in the plane of principal moment. Two 

 important corollaries follow from the theorem just demonstrated, 

 in the case where no forces act : First, the component of an- 

 gular velocity round the axis of primitive impulse is constant 

 during the motion. Secondly, the radius vector which coincides 

 with the axis of G is of constant length during the motion. The 

 first theorem is obvious ; for as the axis of rotation produced by 

 centrifugal force is always perpendicular to the axis of G, it 



