320.] 



BODY WITH FIXED POINT. 



173 



320. The relation between the vectors H and o> is very 

 clearly brought out by making use of the ellipsoids of inertia 

 at the point O. 



The reciprocal ellipsoid at O has the equation (see Arts. 

 269, 270) 



Let P (Fig. 38) be the point where it is met by the vector 

 H\ x, y, z the co-ordinates, p the radius vector of P\ hence 



Fig. 38. 



*/P* y/P> 2 /P 

 H^ = Hy/p, H 



whence 



Fig. 39. 



direction-cosines of H, so that H^ l 

 = Hz/p. The equations (5) give, therefore, 



-.... =tL.iL m= fL.JL 



Mp ' q\* ^ Mp ' q' 3 Mp * qf 



(6) 



where q is the perpendicular let fall from O on the tangent 

 plane at P (see Art. 269). The direction-cosines of CD, 



are the same as those of this perpendicular (ib.). 



It thus appears that the plane through O at right angles to the 

 instantaneous axis 1 is conjugate to the direction of the vector H 

 with respect to the reciprocal ellipsoid at O. 



