1010 Dr. Dorothy Wrinch on the 



to infinity before reaching the line 6 — it, it will approach 

 the line tangentially and pursue the path solely along 

 t|iis line. 



Characteristics of Orbits not restricted to a Sector of 

 the Plane. 



The final case to be considered is when 



hi— 2/<tcos 0i= 2 pm > 2/jl. 



In this case, since 



h 2 = 2/*(m + cos0) 

 and 



V 2 = 2//,(m-f cos 6)jr\ 



the transverse velocity never vanishes in the finite part 

 of the plane. The equation to the orbit in terms of a 

 parameter <b is still given by 



sin 0/2 = ^/ — g— sn (0, ^ —^~ j, 



which is now more conveniently put in the form 



sin 0/2 = sn (^/^4>, V^l)' 



and 



r = 7^ secv2m(^- <fii), 

 U = v /2/*m v^l/ri 2 -l/r- a + Ui72^m. 



The particle goes to infinity, as before, arriving there with 

 the velocity ^/2^m ^/l/r 9 + TJ 1 2 /2fjum, and with ^/2m((f) — 2 ) 

 = 7r/2. Now the equation giving 6 in terms of <£ will show 

 what angular distance from X is described before the 

 particle reaches infinity. It is evident that as m increases 

 this angular distance increases, giving in the limiting case 

 an infinite angular distance. The orbit therefore, in this 

 general case, is a curve described with r and 6 increasing- 

 together and circumscribing the origin an increasing number 

 of times as m increases, and tending to infinity along the 



line = O , p= ( ~~~Jn) • I n the limiting case when m is 



infinite, the path of the carve circumscribes the origin for 

 ever, and only has the circle at infinity as asymptote. 



If the particle is projected w T ith negative or zero radial 



