1008 Dr. Dorothy Wrinch on the 



We have now. therefore, established the existence of a 

 periodic orbit in the field of force 



V= — fiu 2 cos 0, 



viz. the semicircle r = r 1 , in which the periodic time is 

 4? , 1 2 K(l/\/2)/'/yW'. The path of a particle is therefore this 

 periodic orbit when the conditions of projection at r 1 1 



glY Ui = ? r 1 2 V 1 2 = 2 y acos6> 1 , 



and only under these conditions. 



It is further possible to deduce from our general analysis 



that when the transverse velocity of projection satisfies the 



conditions 



0<2 i acos6' 1 -r 1 2 V 1 2 <2 y u, 



this semicircular path is the only periodic orbit possible. 

 We next consider the case when 



O<h 1 a -2ficoa0 1 < 2fi. 

 Putting 



Jl^ — 2/jl cos 6 l — — 2fx cos ol = 2fjbm > 0, 

 we see at once that motion must be restricted to the sector 



-■*< <oc, 

 in which ol>it\2. In this case the (r,t) equation is 

 r = 2(jbm/r 3 , 



glVmg ? = U x 2 -r 2^1^ - 2ftm[i*'. 



If r is originally positive or zero, r increases steadily 

 to the value V (XJ{ 2 + 2/jLm/r 1 2 ) at infinity; while if it is 

 originally negative, it decreases in absolute magnitude 

 until r = r 2 , where 



l/r 2 2 = l/rf + Uffiftm, 



and then changes sign and increases steadily. We may 

 integrate the equation in the form 



t — t l = r 2 (\ / r 1 z -r 2 2 —\/r 2 — r 2 2 )/\/2/j,iu when Ui<0, 



and in the form 



t — t l = r 2 \ / r 2 — r 2 2 /^2/jLm it'U x >0. 



Whatever, then, the initial radial velocity the particle 

 proceeds either directly or after an interval 



2r 2 V ' r{ 2 — r 2 2 \ V2fjnn 

 to increase its distance from the origin at an increasing rate. 



