338 Rotation of a Solid Body round a Fixed Point. 



through which the axis of principal moments shifts in an element 

 of time is = : this angle, multiplied by the constant radius 



vector, will give the elementary motion on the spherical conic 

 traced by the axis of principal moment on the surface of the 

 ellipsoid : this motion is therefore - wQdt ; but in the same time 

 the point of the body which coincides with the point where the 

 axis of moments pierces the spherical conic will describe the 

 angle + u>Qdt in consequence of the angular rotation. Hence 

 the axis of moments will remain fixed in space, and will move 

 in the body with a velocity proportional to the tangent of the 

 angle between the radius vector and perpendicular, the motion 

 being in a direction opposite to the direction of the rotation. 



This is evident from the consideration that Qw = Po> tan 0, Po> 



ft 

 being constant and equal to = (vid. (9)). 



|U.iV 



VIII. To FIND THE MOTION OF THE PRINCIPAL AXIS IN THE 



BODY. 



First Method. 



The point of the principal axis of moments, which is situated 

 at the distance R from the centre, moves on the spherical conic 

 which has been determined. Let this point be projected on the 

 three co-ordinate planes ; then, since the spherical conic is pro- 

 jected into a conic section, the movement of the axis of moments 

 is reduced to the movement of a point on a conic section, accord- 

 ing to a law which must be determined. The radius vector 

 describes an elementary triangle in the surface of the cone (11) : 

 let the projections of this triangle on the co-ordinate planes be 

 (dA lt dA z , dA 3 ) ; we obtain easily 



dAi dz dy dA z dx dz dA 3 dy dx 

 ~dt = y ~dt~ * ~dt' ~dt~ =Z ~dt~ X ~dt ~dt " X ~di ~ y ~dt' 



Substituting in these equations the values of the velocities given 

 by (1), we obtain 



