Orbits in the Field of a Doublet. 995 



The equation may be integrated in the form 



h 2 = 2/jl (cos 6— cos } ) + hi 2 , 



where h takes the value 7i x when takes some specific 

 value 6 1, Now motion can only take place when k is real, 

 so that h 2 must not have negative values. It is therefore 

 evident that no motion can occur when 



2//, cos 6 j— h-f > 2fi, 



and in the critical case, 



2fACOS0 1 -h, 2 = 2fl, 



the motion is restricted to the line # = and is given by 



f = -2^ (3; 



Motion along the axis of the Doublet. 



We may now proceed to discuss these motions along the 

 axis in some detail. 



The integrated form of the equation (1) is 



r 2 = O + 2/i/r 2 . 



If the velocity U at the point of projection r = r l is +Ui, 

 then 



* = U = 4- v'OT+V/f'-sWrT'": 



If JJi 2 >2fijr l 2 , the particle describes the line = 0. in the 

 direction r increasing, with a steadily decreasing velocity. 

 Its velocity at ? j = go is \/~U 1 2 — 2n/r 1 2 . Further, integrating 

 the equation again, the time is given as a function of r in 

 the form 



(_«,=( v/ t U, 2 -2 M /r 1 2 ), 2 + 2 / L t - V-»/(U I 2 -2 /t / n 2 ), 



and the time to infinity is infinite. Thus a particle with 

 these conditions of projection will not leave the system in a 

 finite time. 



The case U 1 2 = 2///r 1 2 gives a similar result. The particle 

 describes the line = in the direction r increasino- with a 

 steadily decreasing velocity, arriving there with zero velocity. 

 The time is given by 



*-*i= (f J -r 1 ?)/2 \/Tfi ; 



so that, as before, an infinite time is required for the 

 description of the line = 0. 



3 S 2 



