288 BOOK OP MATHEMATICAL PROBLEMS. 



prove that the acceleration makes a constant angle with the axis 

 and varies inversely as the cube of the distance from the vertex. 



1383. A cycloid is described under constant acceleration, 

 and 0, <j> are the angles which the directions of motion and of 

 acceleration at any point make with the tangent at the vertex ; 

 prove that 



sin C f<l> A 



^r + log^tan(5-0 

 cos 6 cos (</>- 26) ( \2 



or that <j>-20 = l. 



Zi 



1384. P describes a circle under acceleration tending to S 

 and varying as the distance, S being a point which moves on a 

 fixed diameter initially passing through P; prove that, if 6 be the 

 angle described about the centre in a time t, 



and the distance of S from the centre is =-^ : a being the 



m cos d 6 y 



radius, and in a constant. 



1385. A point describes an arc of a circle, so that its acce- 

 leration is always proportional to the 71 th power of its velocity; 

 prove that the direction of the acceleration touches a certain 



epicycloid, generated by a circle of radius -. - rolling on one 



o 2n 



of radius a - ; a being the radius of the circle described. 



