EXAMPLES. 169 



47. If, at any point of an elliptic orbit about a focus, the force ceases to 

 act for a given very short time, find the angle through which the apse line will 

 have turned and the change of the eccentricity, and show that they are 

 respectively proportional to the resolved parts of the force parallel and 

 perpendicular to the apse line. \ 



48. A particle of mass m describes an ellipse about a focus, pm being the 

 force at unit distance ; when the particle is at an extremity of the minor axis 

 it receives a small impulse mV in a direction perpendicular to the plane of 

 the orbit ; prove that the eccentricity of the orbit will be diminished by 

 ^ V 2 aeln) and that the angle which the axis major of the orbit makes with the 

 distance from the focus will be increased by 



where 2a is the axis major, and e the eccentricity of the orbit. 



49. An ellipse of eccentricity e and latus rectum I is described freely 

 about a focus by a particle of mass m, the angular momentum being mh. A 

 small impulse mu is given to the particle, when at P, in the direction of its 

 motion ; prove that the apse line is turned through an angle proportional to 

 the intercept made by the auxiliary circle of the ellipse on the tangent at P, 

 and which cannot exceed lu/eh. 



50. If the velocity of a periodic comet is suddenly increased near its 

 aphelion by a small amount 8F, prove that the changes produced in the 

 eccentricity and axis major are given by the equations 



where the letters have their usual meanings for elliptic motion. 



51. A comet describes about the Sun an ellipse of eccentricity e nearly 

 equal to unity. At a point where the radius vector makes an angle 6 with 

 the apse line, the comet is instantaneously affected by a planet so that its 

 velocity is increased in the ratio n + 1 : n, where n is great, without altering its 

 direction. Show that if the new orbit is a parabola 



4 



e = 1 cos 2 ^ 6 nearly. 

 



52. A body is revolving in an elliptic orbit with acceleration ^jr 2 to a 

 centre of force in one focus S, and when at P it receives impulsively a small 

 velocity dv in the direction PM at right angles to the major axis. Prove that 



. 8v SM. PM 

 the major axis will turn through the angle -j- ~p . 



53. At a point P of an ellipse, described under a force to a focus S, the 

 direction of motion is deflected through a small angle /3 without alteration of 



