the least weight which will pieaeife the 



f gravity of segment! of 

 i ellipse is a similar concentric ellipse whose 

 minor axis is 



.vf.wher,.* -*(*-. in *). 



: ic foci of a rough prolate spheroid attract 



c; if a |mrticlc without weight be placed on 

 the spheroid, fm-l within what limits it must be placed so aa 

 to be in eqiiilihri'. w that if the coefficient of friction 



be greater than >j -pr, where t is the excenrricity, the 



particle will rest anywhere on the surface. 



20. ilar disc of mass m' and radius c rents in con- 



tact v. , , ( ual unit'. >nn straight rods AB, A \\ un- 



joined at A by a smooth hinge, and which attract each other 



line with a force varying aa the distance; al- 

 disc attracts the rods similarly. Shew that there is equili- 



briu: 



me (2c cos a - a sin a) ma 9 sin 4 a cos , 



where m is the mass of each rod, a the length of each rod, 



and 2a t! nation to eac: 



A square picture hangs in a vertical plane by a string, 



'i passing over a smooth nail has its ends fastened to two 



rrically situated in one aide of the frame. Deter* 



!>oeition8 ot* equilibrium, and whether they are stable 



or unstable. 



I//JT. Let / be tike length of the string, c the distance of 

 th.- two points to which the ends of the string are fastened, 



of a side of the square ; then if /A be gi 

 than r s '(c' + A*) there is only one p*- ium. 



iry position, and the equilibrium ia atoMt/ 

 if M be lest than <?V(c' + A t ) there are two oblique positions 

 of itallt equilibrium, besides the ordinary position of equi- 

 librium, winch is *>&/* with respect to some d 

 and ttfutoofc with respect to other displacements. 



