EXAMPLES. 



A rough cllipt of weight IF can tu 



about one extremity of its major axis, ami t\v v. 

 are suspended by a Btrin.i: which passes over the ]>ully ; wh.-n 



lilihrium its plane is vertical, and its axis inclin- 

 60 to the horizon, prove that th- exevntn. llipse 



is equal to 



V{(3<? + W-P] (Q- W-3P)} 



(C-P)V3 



16. A heavy hemisphere rests with its convex surface on 

 a rough inclined plai. 1 the greatest possible indimi- 

 tion of the plain-. 



17. One end A of a IK aw rod ABC rests again-t a r>ui:h 

 ul plane ; and a point B of the rod is connected with a 



point in the plane by a string, the length of which is - 

 to AB\ determine the position of equilibrium of the md, and 

 shew h<\\- the direction in which the friction acts dcj 

 upon the position of B. 



18. Three equal balls, placed in contact on a h<ri/."iital 

 plane, support a fourth ball. I>eterminc the least values of 

 the coefficients of friction of the balls with each other and 

 with the plane, that the equilibrium may be possible. 



Results. Let W be the weight of each of the three 1 

 balls, W the weiirht of the upper, <f> the angle which the 

 rht line joining tin- cmtre of the upper ball with the 

 of one of the lower balls makes with the vertical ; 



the coefficient of the friction between the balls is tan ^ , and 



the coefficient of the friction between the balls and the plane. 



l|- j 



is jp., w tan ^ . If all four balls are equal we have 



sin </> = -, so that tan * = V3 - \/2. 



19. Determine the curve on the rough surface of an 



very point of which a particle, acted on by three 

 1 forces whose directions are parallel to the axes of the 

 ellipsoid, will rest in a limiting position of equilibrium. 



