70 



Stark, has been given by Epstein and Schwarzschild on Ihe basis of the general 

 theory of conditionally periodic systems which allow of separation of variables. 

 Before we enter on the discussion of the results of the calculations of these authors, 

 we shall first, however, show how the problem may be treated in a simple way 

 by means of the considerations about perturbed periodic sj'stems, developed in § 2. 

 Consider an electron of mass m and charge — e, rotating round a positive 

 nucleus of infinite mass and of charge Ne, and subject to a homogeneous electric field 

 of intensity F, and let us for the present neglect the small effect of the relativity 

 modifications. Using rectangular coordinates, and taking the nucleus as origin and 

 the z-axis parallel to the external field, we get for the potential of the system 

 relative to the external field, omitting an arbitrary constant, 



fi = eFz. 



Calculating now the mean value of .Q over a period a of the undisturbed 

 motion, we see at once, from considerations of symmetry, that this mean value '/' 

 will depend only on the component of the external electric force in the direction 

 of the major axis of the orbit. We have therefore 



^Si 



¥= eF cos^~\ r cos Ifdt, 



where w is the angle between the 2-axis and the major axis, taken in the direction 

 from the nucleus to the aphelium, and where r is the length of the radius-vector 

 from the nucleus to the electron, and ë the angle between this radius-vector and 

 the major axis. By means of the well known equations for a Keplerian motion 



. r COS & = a(cosu + £), — = (1 + e cos uj^r—, 



a It: 



where 2a is the major axis, s the eccentricity and u the socalled eccentric anomaly, 

 this gives 



1 ^ 3 



W= eFcos ^ - - \ a (cos (z -1- c) (1 -}- £ cos u) c?iz ^ ^-saeFcosc;. (74) 



JtTZ \ ^ 



tJo 



We see thus that '/' is equal to the potential energy relative to the external 

 field, which the system would possess, if the electron was placed at a point, situated 

 on the major axis of the ellipse and dividing the distance 2 sa between the foci in 

 the ratio 3:1. This point may be denoted as the "electrical centre" of the orbit. 

 From the approximate constancy of f' during the motion, proved in § 2, it follows 

 therefore in the first place that, with neglect of small quantities of the same order 

 of magnitude as the ratio between the external force and the attraction from the 

 nucleus, the electrical centre will during the perturbations of the 

 orbit remain in a fixed plane perpendicular to the direction of the 

 external force. From the considerations in § 2 it follows further, that the total 



