347-] BODY WITH FIXED POINT. ! 9 ! 



346. The equations of the polhode are easily obtained by 

 considering that this curve is the locus of those points of the 

 ellipsoid whose tangent plane has the constant distance q* from 

 the centre O. Hence, denoting the semi-axes of the momental 

 ellipsoid by a, b, c, the equations of the polhode are 



2 2 .2 2 2 2 



It can, therefore, be regarded as the intersection of the momen- 

 tal ellipsoid with a coaxial ellipsoid whose semi-axes are a*/q', 



Multiplying the second equation by q, and subtracting the 

 result from the first equation, we find the equation of the cone 

 of the body axes 



This is a cone of the second order, concentric and coaxial with 

 the momental ellipsoid. 



The polhode evidently consists of two equal separate branches, 

 of which it is sufficient to consider one. Each branch has four 

 vertices situated in the principal planes of the ellipsoid. 



The herpolhode is confined between two concentric circles 

 whose centre, is the projection of O on the invariable plane. It 

 is a transcendental curve and is in general not closed. 



347. If the momental ellipsoid is an ellipsoid of revolution, the 

 polhode consists of two circles, and the herpolhode is also a circle ; as 

 >' is in this case constant, it follows that <o remains constant. 



If we assume in the general case a > b > c, the polhode reduces to two 

 points whenever q 1 == a or q l = c. The rotation then takes place about 

 a principal axis and is permanent. If q' = b (which does not necessarily 

 mean that the axis of rotation coincides with the middle axis b\ the 

 cone of body axes reduces to two planes 



