308 



24 



From (69) we see that, if we neglect small quantities proportional to F~, there 

 exists a homogeneous linear relation with entire coefficients of the type (13) between 

 the <£>'s, viz. <Wj ^2 — 2(^3 = 0, so that, as far as small quantities proportional to 

 Fare concerned, the mechanical system under consideration appears as degenerate 

 (see page 8) and the motion of the electron can be represented by trigonometric 

 series containing only two fundamental frequencies, for instance co^ and O/;. Of these 

 two frequencies diiîers only little from the frequency of revolution of the electron 

 in a simple Keplerian ellipse corresponding to the motion for F = and for which 

 the values of the /'s are the same, while O/., which is a small quantity proportional 

 to F, may be described as a small frequency which is impressed on the motion of 

 the electron due to the perturbing influence of the external electric field. 



It may be of interest to point out how it can be seen from the formulae (69) 

 and (70) in which manner this small frequency plays a part in the deviations of 

 the motion of the electron from a periodic Keplerian motion. First of all it will be 

 seen that the motion of the electron differs at any moment only by small quantities 

 proportional to F from a Keplerian ellipse with major axis xP. Furtlier, taking 

 mean values, over a time interval extending from t' to t' + ^/<«3, on both sides of 

 the equations (70), we get, denoting the mean values of x, y and z in this time 

 interval by 7y, and C respectively, and neglecting small terms proportional to F, 



C = 1x1(1,-1,), 



$-\-ir^ = ''^xP(i,L,.,e~^^i°F> + c^t^^e^-'OFt), 



where t denotes some moment within the mentioned time interval. Now the quan- 

 tities Ç, 3j and C have a simple meaning. In fact, since the motion which the elec- 

 tron performs in the time interval t' t' i/w, differs from the motion in a Keplerian 

 ellipse with major axis xP only by small quantities proportional to F, the quantities 

 ?, 7} and C may with this approximation be said to represent the coordinates of the 

 mean position of the electron in the Keplerian ellipse which it at any moment may 

 be considered to describe. From symmetry it is seen that this mean position, 

 which may be called the "electrical centre" of the orbit, lies at a point on the 

 major axis, and a simple calculation shows that this point lies at a distance ^/^ea 

 from the nucleus if a denotes the major axis and £ the eccentricity.^) The formulae 

 (71) therefore show that the Keplerian ellipse which the electron at any moment 

 may be considered to describe varies, under the influence of the electric field, its 

 shape and position in such a way that its electrical centre performs an elliptical 

 harmonic vibration in a plane perpendicular to the z-axis round tlie point in which 

 this plane cuts the z-axis. The major axis and the minor axis of the ellipse which 

 the electrical centre describes are equal to 3 z («j + '2 'is) 'i'i3~'2'i3 respec- 



This result follows at once from formula (39) on page 15. Compare also N. Bohr, loc. cit.. 

 Part 11, page 70. 



