Rotation of a Solid Body round a Fixed Point. 339 



dA, 



- a 



. 



X = ( R - a ) P ; 



- ff 



(12) 



r. 



These equations prove that the areolar velocity of the projection 

 on a co-ordinate plane varies as the ordinate to that plane. By 

 means of the method of quadratures, we may determine from 

 equations (12) the position of the projections of the principal 

 axis at any instant, and hence deduce the position of the axis 

 itself. 



Second Method. 



If the spherical conic be projected on a cyclic plane of the 

 ellipsoid of gyration, by lines parallel to x and 2, the projections 

 will be two concentric circles, and the 

 corresponding projections will lie on 

 the same ordinate Sir (fig. 2). The 

 inner circle will belong to the projec- 

 tion parallel to x, if R be greater than 

 6, and will belong to the projection 

 parallel to z if R be less than b ; and 

 if R be equal to , the two circles will 

 coincide with each other and with the Fi g- 2. 



spherical conic, which in this case becomes the circular section of 

 the ellipsoid. The projected point will revolve round the cir- 

 cumference of the inner circle, and will vibrate on the circum- 

 ference of the outer circle between the dotted lines. It is evi- 

 dent that the mean axis of the ellipsoid OY lies in the plane of 

 the figure. Let SI and SI' be equal to /o, />', and let C, C' denote 

 the radii of the two circles : the velocities of the projections in 

 the circles will evidently be 



dt 



