206 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



reaching the highest point. Prove that, if the coefficient of restitution 

 between the circle and the particle is unity, and if the initial velocity is 



the particle after striking the circle will retrace its former path. 



34. A smooth parabolic cylinder is fixed with its generators horizontal 

 and the axis of each of its normal sections is horizontal. A particle is placed 

 upon it at a height above the axial plane equal to the latus rectum ; prove 

 that it will run off at the extremity of the latus rectum, and will then describe 

 a parabola of equal latus rectum. 



35. A particle slides under gravity on a smooth parabola whose axis is 

 not necessarily vertical, and is free to leave it and describe a different parabola 

 under gravity alone. Prove that, if the particle leaves the first parabola at 

 all, it will do so at the point where the normal passes through the intersection 

 of the directrices of the two parabolas. 



36. A particle moves on the outside of a smooth elliptic cylinder whose 

 generators are horizontal, starting from rest on the highest generator, which 

 passes through extremities of major axes of the normal sections. Prove that 

 it will leave the cylinder at a point whose eccentric angle $ is given by the 

 equation 



e 2 cos 3 = 3 cos - 2, 



where e is the eccentricity of the normal sections. 



37. A particle moves in an elliptic tube under a force to a focus equal to 

 pr~ 2 + vr~ 3 . Prove that, if it is projected from the nearer vertex with velocity 



+ e)/a (1 - e)}, the pressure is given by 



&quot;11+ _J_ 11 

 P V 2 2 (l-e) 2 ar} 



38. A particle is constrained to move in an ellipse about a centre of force 

 in one focus varying inversely as the square of the distance, and its initial 

 velocity is such that if it were free its orbit would pass through the other 

 focus. Prove that if the constraint were removed at any point of its path it 

 would describe an orbit passing through the other focus. 



39. A particle is projected horizontally from the lowest point of a smooth 

 elliptic arc, whose major axis 2a is vertical, and moves under gravity along 

 the concave side. Prove that it will leave the curve if the velocity of projection 

 lies between ,J(%ga) and /*J{ga (5 e 2 )}. 



40. A ring is free to move on a smooth elliptic wire whose minor axis is 

 vertical. An elastic thread of natural length I and of modulus equal to n 

 times the weight of the ring passes through the ring and has its extremities 

 fixed to the foci of the wire. Prove that if the ring falls from an extremity 

 of the major axis the pressure at the lowest point will vanish if 



where 2a, 26 are the major and minor axes of the ellipse, and 



