BOOK OF MATHEMATICAL PROBLEMS. 



whi-n /'. Q move up to coincidence with A ; p being tin; radius 

 of cunatmv at J, and * the arc to A measured from u lixrd 

 point. 



1 .>33. A parabola is described about a force in the focus, and 

 alonu' the focal distance SP is measured SQ equal to a constant 

 ;i ; (jl! is drawn perpendicular to the tangent at P to meet 

 the axis in /?; prove that 



<jl! : 2XQ :: velocity at P : velocity at the vertex. 



1334. Prove that the equation 2V* = F.P7is true when a 

 body moves in a resisting medium, F being the extraneous force, 

 and PV the chord of curvature in direction of F. 



1335. Two points P, Q move in the following manner; P 

 de-scribes an ellipse under acceleration tending to the centre, and 

 Q describes relatively to P an ellipse of which P is the centre 

 under acceleration tending to P, and the periodic times of these 

 ellipses are the same; prove that the absolute path of Q is an 

 ellipse concentric with the path of P. 



1336. A particle describes a hyperbola under a force tending 

 to one focus ; prove that the rate at which areas are described by 

 tin- central radius vector is inversely proportional to the distance 

 of the particle from the centre of force. 



1337. A rectangular hyperbola is described by a point undT 

 r-tioH parallel to one of the asymptotes; prove that at a 



point P tin- acceleration is 



/M/ beiuir drawn in direction of the acceleration to meet the other 

 a<\ nij)t"t-, (.' tin- centre, and U the component velocity perpendi- 

 cular to the acceleration. 



133S. A point describes a cycloid under acceleration tending 

 to the centre of the generating circle ; prove that the velocity at 

 any point varies as the radius of curvature. 



1339. A particle constrained to move on an equiangular 

 spiral is attracted to the pole by a force proportional to the 



