I9 2 KINETICS OF A RIGID BODY. [348 



each of which intersects the ellipsoid in an ellipse. These ellipses 

 divide the surface of the ellipsoid into two pairs of opposite regions, one 

 about the greatest axis a, the other about the least c. 



As long as a > <?' > b, the polhode lies in the former region, and the 

 cone of body axes has a as its axis. If b > <?' > c, the polhode lies in 

 the other region, and c is the axis of the cone. 



Two polhodes cannot intersect ; for if they did, the tangent plane a 

 the point of intersection would have two different distances from the 

 centre, which is impossible. 



348. The motion of a body is called stable if after a slight distur 

 bance the body tends to resume the original motion. In our case 

 slight disturbance displaces the instantaneous axis from one polhode to 

 another near by. Hence if the polhode be situated very near to one 

 of the bounding ellipses, the motion is not stable, because a slight dis 

 turbance might change the polhode to one in the other region. The 

 motion is therefore the more stable the more closely the polhode sur 

 rounds either the greatest or the least axis of the ellipsoid. 



If, however, one of the regions between the ellipses be very narrow 

 which will be the case if two of the axes of the ellipsoid are nearly 

 equal, a polhode in this region, though close to the vertex, may stil 

 approach very near to the ellipses so as to make the motion unstable. 



349. Integration of Euler's Equations. As H=o, Euler's 

 equations (4), Art. 328, are 



(4 



ai 



Multiplying by oj, o> 2 , a> 3 , and adding, we find 



whence, by (17), Art 323, 



This is nothing but the equation of kinetic energy. 



