ite 



1012 Dr. Dorothy Wrinch on th 



These orbits are described when and only when the conditions 

 of projection at the point {r^d-i) are such that the radial 

 velocity is zero and the transverse velocity V x is given by 



r*y x * = 2/*cos^. 



Included under these conditions is the case when a particle 



is placed at rest on either of the lines = 77/2, 6= —it /2, at 



any non-zero non-infinite distance r x from the doublet. The 



orbit will then be the appropriate semicircle, and the periodic 



time will be as above. 



(b) When the transverse velocity of projection is such 



that 



< 2//,cos0 l -r 1 2 V 1 2 = 2fim < 2/t, 



the particle only leaves the system if Uj, the radial velocity, 

 is greater than or equal to x/Xfim/r^ The orbit touches the 

 rays 6= +a alternately, where cosa = w, and proceeds to- 

 infinity with a certain line 



6 = O , p = (—dv/d6)u=o, 



as asymptote, if 1^ > *J2fxm\)\. If Ui^ \ / 2jjumjr 1 , the orbit 



touches the rays alternately for ever and does not have a 



linear asymptote. 



If, however, Ui does not attain this critical velocity, the 



orbit touches the rays alternately and keeping to the sector 



\6\ <u, first recedes from the doublet if Ui>0, and then 



approaches the doublet converging on it, through a series 



of waves of decreasing amplitude, which touch the lines 



which bound it. If Ui is, negative, the orbic has similar 



characteristics, the excursion away from the origin being v 



now deleted. However, whatever the value of U l5 under 



the condition 



< 2 A 6COS0 1 -r 1 2 V 1 2 < 2/i 



the orbit is confined between the two rays 0= +«, where 



a, < 7r/2, and 



2\i cos U 1 —lyy ^ = 2[x cos a, 



and is of wave form converging on or diverging from the 

 doublet, the amplitude of the wave varying in such a way 

 that the curve touches the lines which bound it. 



(fc) When the transverse velocity of projection is such 



0< n 2 V 1 2 -2 A 6cos^ 1 < 2fi, 



the orbit, whatever Ui may be, whether it is negative 

 or positive, goes to infinity and the particle describing it 

 recedes from tlio origin — after a finite excursion towards 

 the doublet if Ui is negative — at a steadily increasing rate, 



