Rotation of a Solid Body round a Fixed Point. 343 



IX. To FIND THE POSITION OF THE BODY IN SPACE AT THE 



END OF ANY GIVEN TlME. 



First Method. 



The radius vector of the ellipsoid of gyration, which is per- 

 pendicular to the plane containing the axes of principal moment 

 and of rotation, always lies in the plane of principal moment, 

 and describes in that plane areas proportional to the time. 



Let OGr, Oil be the axes of principal moment and of rota- 



tion ; OR', OO', the axes of centrifugal 

 couple and of corresponding rotation ; 

 the plane CtOQ' will contain the two 

 successive positions of the axis of rota- 

 tion. Let 01 be the position of the 

 axis of rotation at the end of the time 

 $t ; then $u will be equal to the angle 

 described in the fixed plane by the 

 line OR'. Let R' and P f be the radius 

 vector and perpendicular correspond- 

 ing to the centrifugal couple and its 



Fig. 3. 



axis of rotation. The following relations are evident from the 

 figure : 



w sin Q'OI cos 0' 



because 



sinliOI 



^,r\-r 

 sin O 01 = 



, 



sm0 



sin 



sn 



but, from mechanical considerations, 



PR' G 



PRu sin 



; because 



sn 



tP'# 



Hence, by equating the geometrical and mechanical expression, 

 we obtain 



8*. (18) 



