54 



city consider the mass of the nucleus as infinite in the calculations of the pertur- 

 bations of the orbit of the electron. This involves, in the expression for the additional 

 energy of the system, the neglect of small terms of the same order as the product 

 of the intensity of the external forces with the ratio between the mass of the elec- 

 tron and the mass of the nucleus, but due to the smallness of the latter ratio 

 the error introduced by this simplification will be of no importance in the com- 

 parison of the results with the measurements. Since in the case under consideration 

 the system possesses three degrees of freedom, the equations which determine the 

 secular perturbations of the orbit of the electron will correspond to the equations 

 of motion of a system of two degrees of freedom, and it %\ill therefore not be pos- 

 sible to give a general treatment of the problem of the stationary states. Thus, for 

 any given external field, we meet with the question whether the perturbations are 

 conditionally periodic and, if so, in what set ot orbital constants this periodicity 

 may be conveniently expressed. Now, in many spectral problems, the external field 

 possesses axial symmetry round an axis through the nucleus, and in this 

 case it is easily shown that the problem of the fixation of the stationär}' stales 

 allows of a general solution. A choice of orbital constants which is suitable for the 

 discussion of this problem, and which is well known from the astronomical theory 

 of planetary perturbations, is obtained by choosing for (u the total angular momentum 

 of the electron round the nucleus and for «3 the component of this angular mo- 

 mentum round the axis of the field. For the set of ,y's, corresponding to this set of 

 a's, we may take p", equal to the angle, which the major axis makes ^^^th the line 

 in which the plane of the orbit cuts the plane through the nucleus perpendicular 

 to the axis of the field, and ^3 equal to the angle between this line and a fixed 

 direction in the latter plane. For the problem under consideration it will be seen 

 that, with this choice of constants, the mean value 'V of the potential of the per- 

 turbing field will, besides on «j, generally depend on a., and ß., as well as on «3, 

 but due to the symmetry round the axis it will obviously not depend on ^^3. In 

 consequence of this, the equations (46), which determine the secular perturbations, 

 will possess the same form as the Hamiltonian equations of motion for a particle 

 moving in a plane and subject to a central field of force. Thus corresponding to the 

 conservation of angular momentum for central systems, we get in the first place 

 from (46) that a., will remain unaltered during the perturbations. Next corresponding 

 to the simple periodicity of the radial motion in central systems, we see from (46), 

 if ag as well as «^ is considered as a constant, that during the perturbations a^ will 

 be a function of /î, ^nd varj' in a simple periodic way with the time. The per- 

 turbations of the orbit of the electron produced by an external field which pos- 

 sesses axial symmetry will therefore always be of conditionally periodic type, 

 quite independent of the possibility of separation of variables for the perturbed 

 system. As regards the form of the conditiqns which fix the stationary states, it may 

 be noted, however, that with the choice of orbital constants under consideration 

 the /i's will not, as it was assumed for the sake of simplicity in the general dis- 



