On the Motion of a Particle about a Doublet. 381 



If E is negative the particle falls into the doublet after a 

 finite time. The case in which E = is exceptional, and 

 will be considered later. In all other orbits r passes through 

 a single minimum value ? , ( = V / (C/E)), after which the 

 particle passes to infinity, arriving with velocity v ( = ^/(2E)). 

 The changes in r are precisely those which occur when a 

 particle describes a straight line distant r Q from the origin, 

 with a uniform velocity r . 



The remaining integrals are best obtained in terms of a 

 subsidiary variable % given by 



r=r aeox, ..-..,. (0) 



^~°tan % , ,,,,,. (10) 



these equations being mutually consistent with (8). The 

 value of d% is r v o dt\r 2 , and on changing the variable from t 

 to x-> equation (8) yields the integral 



_ c d.e 



X ~J ?i _l M * ™- ft *' *a\ V 



(1+71- cos " TTi cosec- ) 



V Cm JO ) 



which gives a relation of the type 



and on evaluating <b from (4) the integration is complete. 



The special case of E=0 gives motion in a sphere r=r . 

 All the equations of motion are satisfied if r = r together 

 with 



6 L = xcos 6— 



■mr 4 Vq sin 2 6 



Provided n 2 < • there will be two real angles X . 2 



oVom ° 



for which 0, as given by this equation, vanishes ; when iv 

 has the critical value, the angles 6 X 2 coincide in the angle 

 # = tan _1 \/2. There are therefore an infinite number of 

 possible spherical orbits for each value of r , each orbit being 

 confined to a belt of the sphere lying between the cones 1} 2 * 

 Any small departure from spherical motion will be repre- 

 sented by giving to E a value slightly different from zero. 

 For such an orbit it is clear from equation (8) that r ulti- 

 mately becomes infinite or zero, showing that the original 

 orbit was unstable. 



