63-65] APSIDAL ANGLE. 67 



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

 u 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 (j> (u} for f(r\ and a for - , so that 



h z = (f> (a)/a s . 

 Then 



d~x a s d> (a + x) 1 



^ + x + a = 



(j>(a) (a + x) 2 



dffi + 



neglecting x 2 . 



Now if 3 TT^Y i P os itive we ma y P u * ^ equal to /e 2 , and 



then the solution of the above equation is of the form 

 x = A cos (/c0 + a), 



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

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

 justified. 



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



with period 2?r / . / ! 3 ^ . \ \ , the orbit is nearly circular and 

 V I <K<0 J 



its apsidal angle is TT * / J3 -- ~-^ I . 



Again, if 3 ?V\ ^ s ne R a ^i ve we ma j P u ^ ^ equal to /c 2 , 



9 ( a ) 

 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 increases or diminishes, so that x will very 

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



52 



