63-65] APSIDAL ANGLE. 67 



assume that it is always so near to the circle that the difference 

 - - is so small that we may neglect its square, the investigation 



we give will determine under what condition this assumption is 

 justifiable. 



Put u = - + x and write &amp;lt;j&amp;gt; (u] for f(r\ and a for - , so that 



c 



h 2 =(j)(a)/a 3 . 

 Then 



dfa _ a 3 (j&amp;gt; (a + as) 1 



+ ~ 



- 



neglecting a? 2 . 



Now if 3 - -r-y-v- is positive we may put it equal to /c 2 , and 



9 W 

 then the solution of the above equation is of the form 



x A cos (tc6 + a), 



so that the greatest value of x is A t and by taking A small enough 

 x will be as small as we please and the neglect of # 2 will be 

 justified. 



In this case u, and therefore r, will be a periodic function of 6 

 with period 2w /./JS - T/ y [ tn orbit is nearly circular and 



its apsidal angle is TT/ A /J3 - ?M . 

 / V ( 9 a J 



Again, if 3 T/ \ ^ s negative we may put it equal to /c 2 , 

 and then the solution of the above equation is of the form 



and it is clear that one of the terms increases in geometrical 

 progression whether 9 increases or diminishes, so that x will very 

 soon be so great that its square can no longer be neglected, 



52 



