BOOK. OF MATHEMATICAL PROBLEMS. 



1302. If a particle moving under the action of gravity pass 

 through two given points, the locus of the focus of its parabolic 

 path will be a hyperbola. 



1303. A heavy imperfectly elastic particle is projected from 

 a point in a horizontal plane in such a manner that at its highest 

 point it impinges directly on a vertical plane, from which it re- 

 bounds, and, after another rebound from the horizontal plane, 

 returns to the point of projection ; prove that the coefficient of 

 elasticity is L 



1304. If 7*j, r fl , r a be three distances of a projectile from 

 the point of projection, and a,, a 2 , a 3 the angular elevations of 

 these points above the point of projection, 



r, cos 2 a t sin (a a - a a ) + r a cos 2 a a sin (a a - c^) 



f r a cos 8 a a sin (a : a s ) = 0. 



1305. The axis of a parabola is vertical and the vertex 

 downwards, and a circle has its centre at a point P on the para- 

 bola, and passes through the focus S ; a perfectly elastic particle 

 sliding down PS is reflected at the circle and then moves freely 

 under the action of gravity ; find where it next meets the circle, 

 and, if it be at the lowest point, prove that SP is equal to two 

 thirds of the latus rectum. 



1306. A particle is projected up an inclined plane of given 

 inclination so as after leaving the plane to describe a parabola 

 prove that, for different lengths of the plane, the loci of the focus 

 and vertex of the parabolic path are both straight lines. 



1307. A perfectly elastic particle is projected from the middle 

 point of the base of a vertical square towards one of the angles, 

 and after being reflected at the sides containing that angle, falls 

 to the opposite angle ; prove that the space due to the velocity of 

 projection is to a side of the square :: 45 ; 32. 



