258 VII. DYNAMICS OF ROTATING BODIES. 



This vector, being parallel to S e and thus perpendicular to H 2 ' 

 gives, when compounded with J? 2 ' a resultant exactly equal to H 2 . 

 The component of H parallel to co being unchanged by the motion, 

 we find, geometrically, that the angular momentum remains constant 

 throughout the motion, as we have found by a general theorem 

 in 33. 



As we now wish to follow the motion of the body from one 

 instant to another, it will be convenient to free ourselves from the 

 choice of axes which made the instantaneous axis the Z-axis. Let 

 us take for axes the principal axes of the body at 0. Let the com- 

 ponents of the angular velocity ro on the axes be p, q, r. Then the 

 angular momentum, being the resultant of the three angular momenta 

 due to the three angular velocities p, q, r, are by 68, 53) or 

 79, 134), 



19) H x = Ap, H y = Bq, H, = Cr. 



If we draw any radius vector to the ellipsoid of inertia at the 

 fixed center of mass 



the perpendicular d on the tangent plane at the point #, y, g has 

 direction cosines proportional to Ax^ By, Cg. 



If we draw the radius vector p in the direction of the instan- 

 taneous axis, so that 



*>> ! = f = T = f = > 



equations 19) give 



21) H M 



or the angular momentum vector bears to the angular velocity vector 

 the relation, as to direction, of the perpendicular on the tangent 

 plane to the radius vector. Otherwise, if the angular momentum is 

 given, the instantaneous axis is the diameter conjugate to the 

 diametral plane of the ellipsoid perpendicular to the angular momentum. 



The centrifugal couple being per- 

 pendicular to the plane of d and p, 

 lies in the diametral plane conjugate 

 to Q. It produces in the time dt an 

 angular momentum S c dt whose axis 

 is in the same direction. To find 

 the axis of the angular velocity 

 corresponding thereto we must find 

 the diameter conjugate to the plane 



perpendicular to S c , that is the plane yd. But the diameter conjugate 

 to a plane is conjugate to all diameters in it, hence the required 



