996 Dr. Dorothy Wrinch on the 



Now it may easily be seen that -f a /— 2 is the smallest 



velocity o£ projection which causes a particle to recede 

 from the doublet and ultimately to leave the system. When 



the particle returns to the doublet. For since 



r =TJ = S(JJ 1 *+2plr3-2 f i/r*), ... (4) 

 U will vanish at r = r 2 , where 



l/n'-l/V^TV/S/.; 



the particle then reverses its direction and approaches the 

 doublet. Integrating equation (4), we have 



so that the time from r=r l3 at which t — t^, to r==r 2 , at 

 which t = t 2 , is 



the particle reverses its direction, and its velocity is now 

 given by 



and the time therefore by 



t-t 2 = r 2 \/r 2 2 -i J l \ J 2p. 



The time to the origin is consequently r 2 \ v' 2/jl. The particle 

 therefore proceeds in the direction r increasing for a 

 time r 2 s/r 2 2 — 7\ 2 / */2/jl, and then retraces its path, passing 

 through its point of projection with a velocity equal to 

 that with which it was projected and opposite in sign, 

 arriving at the centre of force after a time 



(r 2 2 + r 2 Vr?^ 2 )/ V2ya 



with infinite velocity. 



Considering next the cases when the particle is projected 

 towards the origin from ? , = r 1 with velocity Ui, we obtain 

 the equation 



r = - ^U 1 2 + 2 / ^/r 2 -2 y u7r7, 



and whatever the magnitude of I^ the particle goes to 

 the origin with steadily increasing velocity and arrives 

 there with an infinite velocitv. The times can be obtained 



