BOOK OF MATHEMATICAL PROBLEMS. 



1373. A parabola, y*=2ca;, is described by a point under 

 ration making a constant angle a with the axis, and the 

 velocity when the acceleration is normal is v ; prove that the 

 acceleration at any point (x, y) is 



v'c* 

 (c cos a y sin a) 8 * 



If the point, when the acceleration is normal, be moving towards 

 the vertex, the time in which the direction of motion will turn 

 through a right angle is 



v (sin 3a + sin a) ' 



1374. The intrinsic equation of a curve being s=f((f>), the 

 curve is described by a point with accelerations JT, Y, parallel to 

 the tangent and normal at the point for which <j> = ; prove that 



,-,,. f"4> 



- 3Z ) - sin * S? + 3r ) 



1375. The curve s=f(<j>) is described by a point with con- 

 stant acceleration, which at the origin is in direction of the nor- 

 mal ; prove that its inclination to this direction at any other 

 point is given by the equation 



3 -~'< tan -0 =",. 



1376. A catenary is described by a point under acceleration, 

 whose vertical component is constant (/) ; prove that the hori- 

 zontal component at a point where the tangent makes an angle <j> 

 with the horizon is 



- COS < , 



/ -^-r (1 + m cos < + cos <W- 



1377. A curve is described under constant acceleration paral- 

 lel to a straight line revolving uniformly ; prove that the curve is 

 a prolate, common, or curtate cycloid ; or a circle. 



