KINETICS OF A RIGID BODY. [321. 



321. Again, the equation of the momental ellipsoid at O is 

 (see Art. 266) 



the semi-axes being a = e*/g l , b 



Let the instantaneous axis / meet this ellipsoid at a point P f 

 (Fig. 39) whose co-ordinates and radius vector are x', y\ s', p', 

 so that x'/p', y'/p', z'/p' are the direction-cosines of / and 

 to^wx'/p', Q> 2 = a>y//3 f , (0 3 = (oz'/p'. Substituting these values 

 and introducing the semi-axes a, b, c t we find form (5) 



. . 



whence ~ir + + s = -7- ' ? 



where ^' is the perpendicular let fall from O on the tangent 

 plane at P\ The direction-cosines of H, 



agree with those of q\ 



It follows that the plane of the couple H is conjugate to the 

 direction of the instantaneous axis 1 with respect to the momental 

 ellipsoid at O. 



322. The kinetic energy of a rigid body with a fixed point O 

 has the expression 



where ^mr 2 is the moment of inertia for the instantaneous 

 axis /. 



Now, by Art. 270, the radius of inertia for the line / is equal 

 to the distance of O from the perpendicular tangent plane to 

 the reciprocal ellipsoid, i.e. to q (Fig. 38). Hence 



(8) 



