358 



s of force, 



equilibrium lor the bar. or ton ling as tin- ratio of t lie 



absolute i of the m ore power 1'u 1 lorn-, to that of th- 



powerful, is, or is not, greater than - ; and d: 

 between the stable and unstable ]>< 



7. Two particles connected ly a upport r, 



on the arc of a vertical circle : shew that the centre of gravity 

 is in the vertical through the centre of tlie circle. What is 

 the nature of the equilibrium? 



8. A sphere of radius a, loaded so that the - 

 gravity may be at a given distance b from the cent 

 figure, is placed on a rough plane inclined at an anirle a to 

 the horizon. Shew that there will be two positions of equi- 

 librium, one stable and the other unstable, in which the 

 distances of the point of contact from the centre of gravity 

 are respectively, 



a cos a V(&* a* sin* a), 

 and acosa-fV (&* * sin* a). 



H' nee, find the greatest inclination of the plane which 'will 

 allow the sphere to rest. Is the equilibrium stable or un- 

 jrtable in this limiting case? 



0. A sphere of radius r rests on a concave sph I 

 radius 71; if the sphere be loaded so that the hei-ht of its 

 r.-ntre of gravity fxm the point of contact be ijr, find R so 

 that the equilibrium may be neutral. // = 3r. 



10. A heavy cone rests with the centre of its base on 

 the vertex of a fixed paraboloid of revolution : shew that the 

 equilibrium will he neutral if the height of the n>ne be 



to twice the latus rectum of the genera abola. B 



that the equilibrium is really stal> 



11. A heavy particle attached to one extremity of an elastic 

 string is placed upon a smooth curve, the string lying upon the 



