Orbits in the Field of a Doublet. 1005 



as before, 



and the orbit in terms of a parameter yjr and constants 

 fo and a is 



sin (9/2 = sin (a/2) sn (i/r, sin a/2), 



M = y/u^ — U l 2 /2/>fcCOS a cosh (f — -v/r/V^COSa). 

 It is evident that in this case, as yfr increases from an 

 initial value yjr u u decreases until ^ = £0/ V2 cos a : it then 

 increases to infinity ; while at the same time is going from 

 the value 6 X through the cycle of values (a, 0, —a, 0). A 

 typical orbit of this kind is shown in fig. 3. 



Fio-. 3. 



It is of interest to connect the constants in these equations 

 with the constants given in the initial velocities at the point 

 r i#i« We have, of course, 



V : 2 = 2//,w 1 2 (co9 6 1 — cos a), 



which gives a in terms of i\ 6 Y and Y l . Since O = 1 when 

 r = i\, f is given by the equations 



sin' #i/2 = sin a/2 sn(^ 1 /v / 2 cos a, sin a.j'2) , 



tanh f — xi = ^\\\/%m\\ 

 so that 



g = \/2 cos a sn -1 (sin 6^12 / sin a/2) 



+tanh~ 1 Ui/ v *2jjlui '/cos a. 

 It is perhaps hardly necessary to get the equation in the 

 case of projection towards the origin, since the analysis is 

 entirely similar to that required in the cases when projection 

 is along the axis, and the geometrical characteristics of the 

 curves obtained have already been indicated in the diagram. 

 For example, the part of the curve (fig. 3) in which the 

 particle is approaching the origin is a typical curve of the 

 case when projection is towards the doublet. 



