STATICS. 2G5 



weight is in contact with the whole of the cycloid, the natural 

 length of the string being equal to the circumference of the gener- 

 ating circle ; prove that the coefficient of elasticity is the weight 

 of a portion of the string whose natural length is twice the dia- 

 meter of the generating circle. 



1273. A heavy elastic string, whose natural length is 2, is 

 placed symmetrically on the arc of a smooth vertical cycloid, and 

 when in equilibrium a portion of string, whose natural length is 

 x, hangs vertically at each cusp ; prove that 



2a being the length of the axis of the cycloid, and X the natural 

 length of a portion of the string whose weight is the coefficient of 

 elasticity. 



1274. A heavy elastic string hangs symmetrically over a 

 smooth circular arc, whose plane is vertical, a portion whose natu- 

 ral length is 2a, and stretched length 3, hanging vertically 

 on each side; prove that the natural length of the part in 

 contact is 



4a,/2 log (1 + ^/2), 

 4a being the radius. 



1275. A heavy cone resting symmetrically on a rough sphere 

 may be displaced through an angle of without upsetting, if tin- 



t of the cone be not greater than half a great circle of 

 the sphere. 



