419 



Indeed, if the electron lias not passed th rough the ring, we shall 

 have finally co =z 0, if? =: 0, so that at the end of the encounter the 

 angular velocity of the ring and the velocity of the electron will 

 again have their initial values \)-^, v^. This will also be the case if 

 the electron goes twice through the ring, first in the positive and 

 then in the negative direction. 



If, however, it goes through the ring no more than once, the 

 final value of co will be 4.Tr and according to (12) and (13) the 

 electron will have lost an amount of energy 



ake a^ k* e* 



The ring will have gained just as much. This follows directly 

 from (9) and also from the remark that, as may be seen by (9) 

 and (13), 



^ m u» -h i Q ^' 

 remains constant during the motion. 



In the case &^ = the energy that is imparted to the ring by 



an "effective" encounter is given by 



■ a^k'e* 



(14) 



This agrees with Whittaker's result. In his calculations he has 

 confined himself to a motion of the electron along the axis of the 

 ring, but the preceding considerations show that the theory can 

 easily be generalized. However, it is also seen that, if in an effect- 

 ive encounter the ring is to receive the amount of energy repre- 

 sented by (14), the rotation which may have been imparted to it 

 by a previous encounter, must first have disappeared in one way 

 or another. 



^ 7. If, in the case «>„ = 0, the electron is to pass through the 

 ring for good and all, it must initially have at least the amount of 

 energy (14). If it has less, it can by no means get beyond a point, 

 where 



xp=^mv,' , a) = ~^y^ .... (15) 



ake 



Such a point is really reached, the electron returning after having 

 got to it, when the motion is along the axis. In general, however, 

 the problem is less simple. The locus of the points which satisfy 

 the condition (15) is a surface limited by the circle R and having, 

 for a somewhat high value of v^, the shape of a wide bag lying 

 on the positive side of the circle, which forms its opening. An 



