Orbits in the Field of a Doublet. 1007 



if u=U\ when ^fr = y(r 1 . Thus 



sin 0/2 = ljs/2 sn H^- ^ («-mi)] 

 and sin ft/2 = 1/^2 8.1^. 



The equation connecting and £ can easily be obtained in 

 the form 



In the case of r being negative originally and equal to — Ui, 

 we have 



»•— n = Ui(*i— 0, 



and progress is towards the origin along a curve waving 

 between 0=±7r/2 and cutting the line £ = between each 

 two consecutive grazings of these boundary lines. It may 

 possibly be of interest to remark that the reciprocals of the 

 distances of the various points on the rays 6 = oc, 6 = 0, or 

 0=—a are in arithmetic progression with common difference 

 > /2 /t Kl/v'2/«i. 



A case of special interest occurs, however, when the radial 

 velocity originally is zero. The curve will then be merely 

 the part of the circle r=r x for which 



-tt/2 < < it 12. 



The transverse velocity is given by 



V 2 = 2//, cos 6/r 2 . 



Thus, if a particle is given a transverse velocity 



Vi = yj'ljju cos 6i/ri 



at the point 6 X i\ and no radial velocity, it will describe the 

 semicircle r = i\, \0\ < vr/2 perpetually. 

 The periodic time is easily given by 



* Jo ri V~V^Jo ^costf 



v/2/*Jo y/l- 2 sin 2 6j2 y/tp V ~ } 

 \Z2fjL 



