Rotation of a Solid Body round a Fixed Point. 341 



IT C 



or, since it is easily seen that -= = -^ we obtain finally 



Ji. G 



1 - 7^ sin2 </' 



(14) 



The motion of the principal axis of moments is, therefore, ex- 

 pressed by an elliptic function of the first kind. 



The motion of the axis of moments is determined by the 

 magnitude of the radius vector of the ellipsoid, which is the axis 

 of the original couple impressed upon the body ; if this radius 

 vector be greater than the mean axis of the ellipsoid, the cor- 

 responding spherical conic will have the axis of x for its internal 

 axis ; and if the radius be less than the mean axis, the axis of 

 z will be the internal axis of the conic ; in no case will the mean 

 axis be the internal axis of the spherical conic. If the radius R 

 be nearly equal to either the greatest or least semi-axis, the ex- 

 pression (14) for the time may be integrated. Let R be nearly 

 equal to the greatest semi-axis. The first of the equations (14) 

 belongs to the interior circle, which is of small dimensions in 

 the case supposed ; the second equation expresses the vibratory 

 motion of the projection, through a small arc of the outer circle, 

 which will have a radius much greater than the inner circle ; 

 we may, therefore, suppose the angle i// to be equal to its sine. 



C' 



Multiplying both sides of the equation by -~ we obtain. 



C' , 

 C'K 



C 



Hence 



dt = KM 



0'iL 



--=xm(E't + A). (15) 



If To denote the time of a complete oscillation or revolution of 



