Orbits in the Field of a Doublet. 997 



by integrating the equation in the form 



t - 





LVr f V / W+(U,V~-V)f 2 ' 



thus the time to the origin is 



giving n/2Ui in the case when U\ 2 =2/z,/r 1 2 . The particle 

 can consequently be sent to the origin in any time however 

 small if Ui is sufficiently large, and attain an infinite 

 velocity in the time : it is further evident that it can 

 approach the doublet with an infinite velocity even if 

 it is projected with zero velocity or a velocity away from 

 the origin, however small its 'original distance from the 

 origin, provided only that it is not projected away from 

 the doublet at the point i\ with as great a velocitv as 



As regards motion on the line = it, the equation is 



Thus 



r= O-fy/f*. 



r = + v / U 1 2 -2^/r 2 + L>7r 1 2 



if Ui is the velocit}^ away from the origin at r = r t > 

 Thus r increases as r increases and has the value 



VU 1 2 + 2 / x/?V 



at infinity. The time equation can be obtained as before, 

 and it is found that the time of transit of the line is infinite. 

 If, however, the particle is projected towards the origin 



and vanishes at ?* = r 2 given by 

 the time to r 9 is 



and the particle recedes from the origin and disappears 

 from the system, though only after an infinite time. Thus 

 no conditions of projection in which Ui is finite keep the 

 particle from steadily increasing its distance from the 



