2GO BOOK OF MATHEMATICAL PROBLEMS. 



III. Smooth Bodies under forces in one plane. 



1251. A small smooth heavy ring is capable of sliding on a 

 fine elliptic wire whose major axis is vertical; two strings afc 

 inched to the ring pass through small smooth rings at the foci 

 and sustain given weights ; prove that, if there be equilibrium in 

 any position in which the whole string is not vertical, there will 

 be equilibrium in every position. 



Prove also that the pressure on the curve will be a maximum 

 or minimum when the sliding ring is at either extremity of the 

 major axis, and when its focal distances have between them the 

 same ratio as the two sustained weights. 



1252. Two spheres of densities p, <r and radii a, 5, rest in a 

 paraboloid whose axis is vertical, and touch each other at the 

 focus; prove that pV = <r 8 & 10 . Also if W, W" be their weights, 

 and 7?, R the pressures on the paraboloid at the points of 

 contact, 



_ 7? _1/_.7A 

 W JF'~2\FT WJ' 



1 253. Four uniform rods freely jointed at their extremities 

 form a parallelogram, and at the middle points of the rods are 

 small smooth rings joined by rigid rods without weight. The 

 parallelogram is suspended freely from one of its angular points; 

 find the tensions of the rods and the reactions of the rings, and 

 prove that (1) if the parallelogram be a rectangle the tensions 

 are equal, (2) if a rhombus the reactions are equal. 



1254. An elliptic lamina of axes 2a, 26, rests with its plane 

 vertical on two smooth pegs in the same horizontal line at a 

 distance c; prove that, if c < bJ2 or > c^/2, *the only positions 



