417 



of a "liiie", and we shall denote by a ihe radins and by k the 

 amount of inagn^isin per unit of length. Let the centre be taken 

 as origin of coordinates, the axes Y and Z, being in the plane 

 of the circle, and let .v be the distance from a fixed point, measured 

 along the circle. The positive direction of s will be determined by 

 the rotation O Y -^ Z, and will therefore correspond, as we 

 shall say, to the dii-ection of U X. We shall finally suppose the 

 ring to be a rigid body that can only rotate about X, and we 

 shall in the first place calculate the couple acting on it when an 

 electron with charge e moves in the neighbourhood. 



The force on an element ds is /cgds and its moment with 

 respect to X a k gs (f^ = '^ k hs ds. Thus the resultant couple is 



ak I hgds, where the value of the integral may be deduced from 



(3). For this purpose we imagine some stationary surface o having 

 the circle R for its boundary and the normal n to which is diawn 

 in a direction corresponding to the positive direction of s. Then, if 

 this surface does not intersect the electron, 



I hs ds = i dindO=: — j j dn do' 



(7) 



We shall suppose the motion of the electron to be so slow and 



to change so slowly that it may be said, in any of its positions P, 



to be surrounded by the electric field that would exist if the electron 



were at rest in that position. Then the last integral in (7) has the value 



e 

 — to, if to is the solid angle subtended at F by the ring R, the 



4:71 



sign of to depending on the direction, towards the positive or the 

 negative side, in which straight lines drawn from F pass through 

 the surface. Hence, the equation of motion of the ring will be (^ 

 angular velocity, Q moment of inertia) 



dd- ake da> 



dt 4: jr c dt 



If this equation is to hold for a certain lapse of time, the surface 

 a must be chosen in such a way as not to be traversed by the 

 electron during that iiiterval. 



Now, two cases must be distinguished, the electron passing or 

 not passing across the circular plane within the ring, or, as we 

 shall say, through the ring. In the latter case, o may be made to 

 coincide with the circular plane and we shall have, both before 

 and after the encounter, if the electron is at a great distance, 



27* 



