I4 6 KINETICS OF A RIGID BODY. [267. 



Dividing (n f ) by the square of the radius vector, /a 2 , and com- 

 paring with (12'), we find 



e 2 e 2 

 ?=->/>=-> (16) 



as is otherwise apparent from the fundamental property of the 

 momental ellipsoid (Art. 259). 



The equation of the sphere of radius q described about O as 

 centre, ^ 2 +j 2 +^ 2 =^ 2 , together with (ii f ), represents the curve 

 of intersection of the ellipsoid with the sphere. Through this 

 sphero-conic passes the equimomental cone, all of whose lines 

 have the moment of inertia I=Mq\ Hence, the equation of 

 this cone can be written in the form 



267. If we assume I I > 7 2 > 7 3 , and hence q\>q^> q& q must 

 be < e 2 /^ and >e 2 /^ 1 . As long as q is less than the middle 

 semi-axis e 2 /^ 2 of the ellipsoid, the axis of the cone coincides 

 with the axis of x, but when q>^/q^ the axis of z is the axis 

 of the cone. For = e 2 /^ 2 the cone degenerates into the pair 

 of planes (q\qf)x' L (q^qf]z^=Q. These are the planes of 

 the central circular (or cyclic) sections of the ellipsoid ; they 

 divide the ellipsoid into four wedges, of which one pair contains 

 all the equimomental cones whose axes coincide with the great- 

 est axis of the ellipsoid, while the other pair contains all those 

 whose axes lie along the least axis of the ellipsoid. 



268. There is another ellipsoid closely connected with the 

 theory of principal axes ; it is obtained from the momental 

 ellipsoid by the process of reciprocation. 



About any point O (Fig. 32) taken as centre let us describe 

 a sphere of radius e, and construct for every point P its 

 polar plane TT with regard to the sphere. If P describe 

 any surface, the plane TT will envelop another surface-. which is 



