350 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



244. A uniform chain hangs in equilibrium with its ends fastened to two 

 points nearly in the same vertical line. One end is released, and when the 

 velocity of this end is u the lengths of the two portions of the chain are I 

 and I'. Prove that, if the other end is released at this instant, the chain 

 becomes straight after a time 1(1 + 1l'*}\l'u. 



245. A uniform chain is suspended from two points in the same horizontal 

 line so that the tangents at the ends make angles a with the horizontal. Prove 

 that, if the support at one end is removed, that end starts to move in a 

 direction making with the horizontal an angle 6, where 



tan 6 = ( 1 + sin 2 a + 2a tan a)/sin a cos a, 



and that the tension at the other end is diminished in the ratio 



1 : 1 +i<z- 1 cota. 



246. A chain is attached to two fixed points and rests in an arc of a 

 circle of angle 2a under a repulsive force varying inversely as the cube of 

 the distance from the extremity of the diameter bisecting the arc. Prove 

 that, if the chain is cut through at its middle point, the initial tension at 

 any point is proportional to 



sec 2 10 sech a cosh (a - <), 

 where is the angular distance of the point from the point of section. 



247. A heterogeneous chain hangs under gravity in the form of a circle, 

 its ends being free to slide on two smooth straight wires which make equal 

 angles y with the vertical. Prove that, if the chain is severed at its vertex, 

 the tension at a point where the tangent makes an angle $ with the horizontal 

 is diminished in the ratio 



< : y + cot y. 



248. A chain of variable density has its ends fixed at points A, B and 

 hangs freely, the tangents at A and B making angles a and /3 with the hori- 

 zontal. Prove that, if the end A is released, the tension at a point P, where 

 the tangent makes an angle <p with the horizontal, is instantaneously 

 changed in the ratio 



(4>+a)sin/3 : cos (3 + (a + /3) sin /3. 



