15 



299 



.r I I y 



where to is the trequency of revolution of the electron in its Keplerian orbit, and 

 where d is an arbitrary constant. The last expression is easily seen to be identical 

 with the expressions for the coordinates of a point performing a Keplerian motion, 

 which are well known in celestial mechanics'), and from which, as mentioned in 

 the beginning of this section, the expression (37) could have been deduced directly. 



In the preceding considerations the problem has been treated as the 

 problem of the motion of the electron in a plane. If we, however, consider the 

 molion of the electron in space, we have to do with a mechanical system of three 

 degrees of freedom. This system will appear as a degenerate system, because there 

 will occur in the trigonometric series representing the displacement of the electron 

 in any direction in space only two fundamental frequencies, viz. the frequency 

 of the radial and the mean frequency co^ of the angular motion of the electron in 

 the plane of its orbit. In the presence of a homogeneous magnetic field, 

 however, the system will no more be degenerate, because a third fundamental 

 frequency will occur in the motion of the electron, which no longer will remain plane. 

 In fact, assuming that the intensity of the magnetic force is so small that we may 

 neglect small quantities proportional to the scjuare of this intensity, we have accor- 

 ding to a well known theorem of Laumoh, that every possible motion in the pre- 

 sence of the magnetic field may be obtained by superposing on a possible motion 

 of the system without field a slow uniforin rotation round an axis through the 

 nucleus which is parallel to the direction of the field. The fre(|uency of this rota- 

 tion will be given by 



where c is the velocity of light and H the intensity of the magnetic force. From 

 this we see that the mean frequency of rotation of the electron round the above 

 mentioned axis, which we will denote by w.,, will be equal to w.j iu.,Az^ih where 

 the upper or lower sign holds according to whether the direction of the superposed 

 rotation has the same direction as or the opposite of that of the rotation of the 

 electron round this axis. 



Let us now ask for the trigonometric series in which the (lisplacenicnl i)t the 

 electron in dilTerent directions in s])ace can be expanded in the jjresence of a 

 magnetic field. Take the nucleus as origin of a system of rectangular Cartesian 

 coordinates x, y, z, the z-axis of which is parallel to the direction of the magnetic 

 field. Let the angle between the z-axis and the plane in which the electron at any 

 moment moves be denoted by and let the position of the electron in this plane 

 be described by means of rectangular coordinates ç, r^, the ;^-axis being perpendi- 



(40) 



M See loi- instance (.hahlikh. Ioc cit. 1, p. 21.'>. 



