294 BOOK OF MATHEMATICAL PROBLEM*. 



1410. A particle is acted on by a repulsive force tending from 

 a fixi-d point, and by another force in a fixed direction ; when at 

 a distance r from the fixed point, the accelerationa of these 

 forces are 



r a ( l ~a)' r*(c 2+ J 



respectively ; prove that the particle, abandoned motionless to 

 these forces at any point where they are equal, will proceed 

 to describe a parabola, of which the fixed point is focus. 



1411. A bead moves on a smooth elliptic wire, and is at- 

 tached to the foci by two similar elastic strings, the natural 

 lengths of which are equal, and which remain stretched through- 

 out the motion ; prove that, if projected with proper velocity, the 

 velocity will always vary as the conjugate diameter. 



1412. A force /resides in the centre of a rough circular arc 

 and from a point of the circle a particle is projected with a 

 velocity v(Jaf) along the interior of the circle; prove that 

 the normal pressure on the curve will be diminished one half after 

 a time 



a being the radius and n the coefficient of friction. 



1413. A heavy particle is projected inside a smooth parabo- 

 loid of revolution, whose vertex is its lowest point, and its great- 

 est and least vertical heights above the vertex are h y h' ; prove 

 that, if v, v be the velocities at these points, 



and that the pressures on the paraboloid at the corresponding 

 points are inversely as the curvatures of the generating parabolas 

 at these points. 



1414. A particle is projected horizontally so as to move on 

 the interior of a smooth hollow sphere of radius a, and the velo- 



