DYNAMICS OF A POINT. 305 



1 i />6. A point describes half the arc of a cardioid, oscillating 

 symmetrically about the vertex, in such away that the hodograph 

 is a circle, the origin being in the circumference; prove that the 

 acceleration of the point describing the cardioid will vary as 

 -/ - 3a; r being the distance from the cusp, and 2a the length of 

 the axis. 



1457. A point P describes a catenary in such a manner that 

 tight line drawn from a fixed point parallel and proportional 



to the velocity of P sweeps out equal areas in equal times ; prove 

 that the direction of fa acceleration makes with the normal at P 



an angle 



/2 

 tan' 1 r^ 



< being the angle which the normal makes with the axis. 



1458. If a circle be described under a constant acceleration 

 not tending to the centre, the hodograph is a lemniscate. 



1459. A curve is described under constant acceleration, and 

 its hodograph is a parabola in which the radii vectores are drawn 

 from the focus; prove that the intrinsic equation of the curve is 



ds ><t> 



</<r csec f- 



14GO. A point describes a curve whose hodograph is a circle 

 described with uniform velocity about a point in tin- cireumtcr- 

 ence; prove that the curve is a cycloid described with con.-tant 



acceleration. 



14C1. A point de-'i-ibes a certain .m-vo, its acceleration 1 

 initially normal; and \vln-n its direction of motion has turned 



_'ii an angle <, that of acceleration has turned tlmm- 

 angle 2< in the same sense; prove that the hodograph of the 

 path is a circle described about a point, in the cireu: 



14C2. A particle is constrained to move in an elliptic t 

 und'-r tw.i foroei to tli.- vin;; inversely as the square 



of the distance and e.jual at njual distances, a- ;>laced 



un -table e<|uilibriuni ; prove that the ! 

 graph is a cir 



w. 20 



