BOOK OF MATHEMATICAL PROBLEMS. 



1419. A heavy particle is attached to a fixed point by a fine 

 string of length a, and when the string is horizontal and at its 

 fill length, the particle is projected horizontally at right angles to 

 the i-tring with a velocity due to a height 2a cot a; prove that the 



greatest depth to which it will fall is a tan -. 



1420. Two heavy particles are placed on a smooth cycloidal 

 arc whose axis is vertical, and are connected by a fine string 

 passing along the arc : c is the distance of either particle mea- 

 sured along the arc from its position of equilibrium ; prove that 

 the time of arriving at a distance s from the position of equi- 

 librium is 



y. log . 

 g c 



a being the radius of the generating circle. 



1421. An elliptic wire is placed with its minor axis vertical, 

 and on it slides a smooth ring to which are attached strings, 

 passing through fixed rings at the foci and sustaining each a 

 particle of weight equal to that of the ring : determine the velo- 

 city which the particle must have at the highest point that the 

 velocity at the lowest point may be equal to that at the extremity 

 of the major axis. 



1422. Two particles of masses p, q are connected by a fine 

 string passing through a small fixed ring ; p hangs vertically, and 

 q is held so that the adjacent part of the string is horizontal: if 



q be let go, the initial tension of the string is - , and the 

 initial radius of curvature of y's path is 



a 



j 



a being the initial distance of q from the ring. 



1423. Two particles of masses m, m lying on a smooth hori- 

 zontal table are connected by an inextensible string at its full 



