. Orbits in the Field of a Doublet. 1003 



or writing f — ? =& 



sin 0/2 = sin (a/2) sn (%/\/2 cos a, sin a/2), 

 i£ = \/Ui 2 /2ft cos a — u x 2 sinh (f — %) . 



This curve consists of a series of waves touching 8= + a 

 alternately and going to infinity along the line 



x=fc r-(-i)„,-"- 



given by 



sin o /2 = sin a/2 sn [f / \/2 cos a, sin a/2] . 



To obtain £ in terms of the velocities of projection UiVj 

 and the coordinates of the point of the projection r x 0i, 

 we may remark that, if % = %], when = 6 X , 



sin A /2 = sin a/2 sn (^ / \/2 cos a, sin «/2) 

 tanh f — %i = ^i v/ 2yct cos «/Ui. 



Fig.l. 



Thus 



£ = V2 cos a sn -1 (sin 0j/2 / sin a/2) 



4- tanh -1 ^ V'lfi cos a/Ui ; 

 and hence the direction = is given by 



sin o /2 = sin a/2 sn ["sn" 1 (sin ^/2 / sin a/2) 



H - tanh -1 z/ T \/2yLt cos a / Ui 



V 2 cos a J 



A typical curve of this class is shown in fig. 1. 



