MI n:\i. i\'rn.iHi:ir.M. 



quired to determine whether the equilibrium 

 Or unsta: 



Since the equilibrium is neutral, tlie centre of gravity Q 

 must coincide with the centre 

 of curvature of tin; pMierar 

 parabola at the vertex; m>\v, if 

 el ill', rent points be taken in a 

 parabola, the further the assumed 

 point is from the vertex, the 

 further is the point of inter 

 tion of the normal ami the. axis 



from the vertex. Hence, the 



normal AN in the figure meets 



the axis of the parabola further from a than G is, and the 



equilibrium is stable. 



It is easy to shew generally, that if a portion of a 

 of revolution rest in neutral equilibrium with its vertex on 

 a horizontal plane, the equilibrium is really stal)le or unstuM-, 

 according as the radius of curvature of the generating curve 

 has a minimum or maximum value at the vertex. 



266. The results of Art. 264, when the sections BAC and 

 DAE are circles, may also be obtained by using the theorem 

 which we have quoted in Art. 262. 



Let z denote the height of the centre of gravity g above 

 the horizontal line through 0, and let Ng = c ; t 



Expand the cosines in powers of the angles ; thus 



.{.(,+)-+};+. 



