Professor ^Iac Cullagh's Lectures on Rotation. 143 



cos j3 cos 7 (b- — C-), cos a cos 7 (e- — a-), cos a cos /3 (a^ — 6^) ; 



these three expressions multiplied by fiur will produce the quantities used in 

 (G). Hence it appears, that the couple produced by the centrifugal forces lies 

 in tlie plane of tlic radius vector and perpendicular to a tangent plane of the 

 ellipsoid (4); the tangent plane being perpendicular to the axis of rotation. Also, 

 the magnitude of the resultant couple is proportional to the triangle formed by 

 the radius vector and perpendicular. 



The diflerential equations of motion commonly used in the solution of this 

 problem may be deduced immediately from equations (6). In fact, as the axes 

 of co-ordinates are axes of permanent rotation, the increment of angular velocity 

 round each axis will bo equal to the statical couple of the applied forces (in- 

 cluding centrifugal forces), divided by the moment of inertia round that axis ; 

 tiie statement of this fact, in analytical language, will give the equations of 

 motion : 



B^^^{(J-A)pr + M■, (7) 



(L, M, K) being the components of the applied statical couple. 



The position and magnitude of the couple produced by the centrifugal 

 forces are easily found by the method which has been just given; but the 

 direction will be found more readily by taking more particular axes of co-ordi- 

 nates. Let the axis of rotation be the axis OZ, and 

 the plane of radius vector and perpendicular be the 

 co-ordinate plane XOZ. In the accompanying figure 

 OR' and OP' are the radius vector and perpendicular 

 of the ellipsoid (2), and OR, OP the radius vector and ^ 

 perpendicular of the ellipsoid (4), which is reciprocal 

 to the former ; the rotation is positive, in the direction 

 indicated by the arrow. As the rotation is round the 

 axis of 2, it is easy to see that the statical couple produced by centrifugal force 





