326 



36 



where Ihe summations are to be extended over all positive and negative entire 

 values of r except r = 0, and where r\ s, s', n and // have the same signification 

 as in (94). 



In the special case where the undisturbed orbit is circular, i. e. c' = 1, s = 0, 

 the formulae (94) and (96) assume the simple form 



, SeFx'-P' '^ ., . 



From this it is seen that in this case the frequency , which did not appear in 

 the undisturbed motion, has also disappeared from the perturbed motion. Moreover 

 we learn from (97) that the perturbed motion, with the approximation involved in 

 this formula, takes place in a plane which rotates uniformly round the z-axis with 

 a frequency (— r 0^3). In this plane the perturbed motion is periodic with period 

 ^2 and may be represented by 



^ + 1, = - e^'"--' + -^^^.(3-e^--"'-'), 



wehere the v^-axis is perpendicular to the z-axis. The "centre of gravity" of the per- 

 turbed orbit is seen to be displaced, under the influence of the electric field, to a 



point on the c-axis situated at a distance ^^^'^ ^ 3,^ from the nucleus, in a direc- 

 ^ 4o ' 



tion opposite to the direction of the component parallel to the ?-axis of the force 

 which the electric field exerts on the electron. 



Another case in which the perturbed motion assumes a simple character is 

 that for which the plane of the undisturbed orbit is perpendicular to the direction 

 of the electric field (//' = 1, //. = 0). In this case (94) and (96) assume the form 



r' = ^"^^ [-^-L J~(.^_,(rc)-./r+i(r£l) cos 2;rrrt;,/j, .r' ^ /{/' - O. j (98) 



We see that the frequency w.^ does not appear in the perturbed motion, and from a 

 comparison of (98) and (90) we learn that this motion may be described as a Kep- 

 lerian motion of frequency w, in a plane which makes a small angle equal to 

 4o'^ -—7 with the x-y-plane, and which rotates with a frequency —Wj-^-w,^ 

 round the z-axis. The minor axis of the Keplerian ellipse is at any moment paral- 

 lel to the æ-j/-plane, and the direction in which the electrical centre is "pushed" 

 out of the .r-i/-plane coincides with the direction of the force which lh(^ ])erturbing 



