Orbits in the Field of a Doublet. 1001 



the doublet, however great its initial radial velocity towards 

 the. doublet may be. If, however, the angular momentum 

 is not great enough to enable the particle to get beyond 

 the lines | 6 | ==7r/2, or is great enough to enable it to regain 

 the sector \0\<_ tt/2, it is only in the case of a sufficiently 

 large radial velocity away from the origin that the particle 

 can be prevented from approaching the centre (possibly 

 after an excursion away from it finite in time and extent) 

 and bombarding the doublet with an infinite momentum 

 after a finite time. 



Characteristics of the Orbits which lie in a Sector 

 of the Plane. 



We may now discuss the variation in V, the transverse 

 velocity. 



Suppose, first, that we limit ourselves to the cases when 



< cos 0i~ r l 2 Yi 2 /2/jb = cos a. = m < 1, 



in which the motion is restricted between the lines 6=±cc 

 and a. < 7r/2. 



V is always zero at all points of these bounding rays. 

 Hence, except when U the radial velocity also happens 

 to vanish at a point on the bounding rays, at such a point 

 the velocity of the particle is entirely along the radius 

 vector. Consequently at such a point the orbit touches 

 the bounding rays, except in the special case when the 

 radial velocity is also momentarily evanescent. 



Motion therefore takes place in general along a curve of 

 wave form touching alternately the bounding rays 0= +a. 

 If the particle is projected at r x Q^ with velocities U = Ui > 0, 

 V = V 1 , the particle initially recedes from the doublet, 

 touching the bounding rays alternately. If 



Ui> S (2 fi cos ufa*) 



it gets perpetualh r further from the origin. If, however, 



V x < V (2/x cos oLJr^) 



the particle recedes from the origin along a curve of 

 wave form, touching the rays alternately until it reaches 

 a point r 2 given by 



l/jY-l/r 2 2 = Ui 2 /2/acos«. 



At this point its velocity along the radius vector is zero, 

 and it therefore touches the circle r=r 2 . Subsequently 



