BOOK OF MATHEMATICAL PROBLEMS. 



1 :'.L'J. A heavy particle is projected so as to move on a ciicu- 

 lar arc whose plane is vertical, and afterwards describe a paraliola 

 irerlv ; prove that the locus of the focus of the parabolic path is 



an epicycloid formed by a circle of radius . rolling on a circle of 

 radius * ; a being the radius of the given circle. 



1323. A cycloidal arc is placed with its axis vertical and 

 vertex upwards, and a heavy particle is projected from the cusp up 

 the concave side of the curve with velocity due to a height k ; 

 prove that the latus rectum of the parabola described after leaving 



the curve is ^-, where a is the length of the axis of the cycloid. 



1324. If a cycloidal arc be placed with its axis vertical and 

 vertex upwards, and a heavy particle be projected from the cusp 

 up the concave side of the curve ; the focus of the parabola 

 described by the particle after leaving the curve will lie on a 

 fixed cycloid of half the dimensions. 



1325. In a certain curve the vertical ordinate of any point 

 bears to the vertical chord of curvature at that point the constant 

 ratio 1 : m, and a particle is projected, from the point where the 

 tangent is vertical, along the curve with any velocity; prove that 

 the height ascended before leaving the curve : height due to 

 velocity of projection :: 4 : 4-f w. 



1326. A smooth heavy particle is projected from the lowest 

 point of a vertical circular arc with a velocity due to a space 

 equal in length to the diameter 2a : the length of the arc is 

 such that the range of the particle on the horizontal plane through 

 the point of projection is the greatest possible ; prove that this 

 range is equal to a ^/(9 + 6 



