EXAMPLES. 171 



59. In the last Example there is no resistance but there is a disturbance 

 which produces a normal acceleration g. Show that the maxima of the rates 

 of variation of the principal semi-axes of the instantaneous ellipse are given 

 by the equations 



a = f&amp;gt; = 9 

 b a (a + fyfjfj. 1 



where /A is the central force on unit mass at unit distance. 



60. A particle P describes an ellipse under a central force producing an, 

 acceleration k* (distance) directed to a point 0. When P is at an end of the 

 axis major, begins to move with a simple harmonic motion psmXt. Show 

 that the motion of P may be represented at any time by motion in an ellipse 

 whose centre is fixed and axis minor is constant and whose semi-axis major is 

 variable according to the equation 



a = a + ^ , 2 (X sin kt-k sin \t] sec kt. 



