1002 Dr. Dorothy Wrinch on the 



it approaches the doublet, converging on it along a wave- 

 like path. In the case of an orbit of: this type, the particle 

 approaches the doublet with infinite velocity. 



The equations of the orbits in these cases can be obtained 

 either by integrating the original equation (1), which with 

 the help of equation (2) may be written 



2 (cos — cos a) d 2 u/d6 2 -sin 6 dujdd- 2u cos a = 

 or 



\/(cos# — cos a) -77; ( \/(cos 6 — cos a) -tq ) — wcos a = 0, 



or we may obtain them by using the fact that XJ/V = dr/rdd. 

 Proceeding from this, we obtain 



-dujude = drfrcte = r/rd = + •(^J«i , + -U.V>«»-jO. 



1 11— u \/ (cos V J cos u — I) 



Hence 



du dd 



V^Uj 2 + 2//V - '/fLUj 2 cos a y/ty \/ cos - cos a 



Now if the orbit is being described initially with r and 6 

 increasing, U/V>0. This case needs the lower sign in the 

 above equation. 



sin 0/2 — sin a/2 sn (yfr, sin a/2), 



where sn is the Jacobian elliptic function with modulus 

 sin a/ 2, and we obtain 



_ d0 _ _ = \/l d ± 



\/ 2 Py/ cos 6 — cos a. \/2fx 



Further, putting 



u = \/JJ 1 *j2fjL cos ol — ux sinh f, 

 we have 



\/TJ i 2 + 2fjLU 2 —2/iiu 1 2 cos a V^ cos a 

 Hence £ & 



- - V2f , 



Vcos a 



where £ * s some constant ; and we obtain the equation 

 of the orbit in terms of a parameter f and constants a, £ 

 in the form 



sin 0\2 = sin (a/2) sn [(f -f)A/2 eos a, sin a/2], 



w = <v/(Ui 2 /2/<tcos a — i^ 2 ) sinh f, 



