1000 Dr. Dorothy Wrinch on the 



Characteristics of the Radial Motion : Reduction to the 

 Case of Motion on the Axis of the Doublet. 



Consider next the (r, t) equation relating the radius 

 vector to the time. We have 



r ~r6 2 - r-h 2 /r 3 = -2jl cos 6/rK 



Consequently, 



r = — 2/^m/r 3 , 

 where 



m = cos^ — h^/2/jb 



as before. Now, we have already discussed the equations 



r — — 2/a/V 3 , f = + 2/ju/r z , 



which give the motion on the lines = and 6 = it re- 

 spectively in the case when the particle has no angular 

 momentum. The results in two cases can be readily 

 modified so as to cover the general case of motion — the 

 first equation for the case when m is positive, and the 

 second for the case when m is negative — by merely writing 

 2jjm for 2/x. In fact, so far as the behaviour of r is 

 concerned, the characteristics of orbits in which 



>r 1 2 V 1 2 /2^-cos(9 1 > -1 



are similar to those on the line 6 = ; and the characteristics 

 of orbits in which 



r 1 2 Y l 2 /2fi-cos0 1 > 



are similar to those on the line 6 = tt. 



Thus, in the first set of cases, no conditions of projection 

 with respect to the radial velocity can prevent the particle, 

 either immediate!}' or after a finite interval, receding from 

 the origin with an ever-in creasing velocity. In the second 

 set of cases, it is only when the radial velocity is sufficiently 

 big, viz. , 



that the particle leaves the system. In all other cases 

 the particle arrives at the doublet in a finite time with 

 an infinite velocity. 



The difference in the characteristics of these two classes 

 of orbits is of course not unexpected. For when | 6\ >ir/2 

 the radial force is repulsive, and for \6\ <tt/2 attractive, 

 On |0|=7r/2, it is zero. Consequently, if the particle 

 has sufficient angular momentum to pass the line |0|=7r/2, 

 and not sufficient to regain the sector | | <. 7r/2 after one or 

 more revolutions, nothing can prevent it from receding from 



