represents the frequency of rotation of the major axis.
Before proceeding further, it might be of interest to note that this fixation of the stationary states of the hydrogen atom perturbed by external electric and magnetic forces does not coincide in certain respects with the theories of Sommerfeld, Epstein and Debye. According to the theory of conditionally periodic systems the stationary states for a system of three degrees of freedom will in general be determined by three conditions, and therefore in these theories each state is characterized by three whole numbers. This would mean that the stationary states of the perturbed hydrogen atom corresponding to a certain stationary state of the unperturbed hydrogen atom, fixed by one condition, should be subject to two further conditions and should therefore be characterized by two new whole numbers in addition to the number
. But the perturbations of the Keplerian motion are simply periodic and the energy of the perturbed atom will therefore be fixed completely by one additional condition. The introduction of a second condition will add nothing further to the explanation of the phenomenon, since with the appearance of new perturbing forces, even if these are too small noticeably to affect the observed Zeeman and Stark effects, the forms of motion characterized by such a condition may be entirely changed. This is completely analogous to the fact that the hydrogen spectrum as it is usually observed is not noticeably affected by small forces, even when they are large enough to produce a great change in the form and position of the orbit of the electron.
Relativity effect on hydrogen lines. Before leaving the hydrogen spectrum I shall consider briefly the effect of the variation of the mass of the electron with its velocity. In the preceding sections I have described how external fields of force split up the hydrogen lines into several components, but it should be noticed that these results are only accurate when the perturbations are large in comparison with the small deviations from a pure Keplerian motion due to the variation of the mass of the electron with its velocity. When the variation of the mass is taken into account the motion of the unperturbed atom will not be exactly periodic. Instead we obtain a motion of precisely the same kind as that occurring in the hydrogen atom perturbed by a small central field. According to the correspondence principle an intimate connection is to be expected between the frequency of revolution of the major axis of the orbit and the difference of the frequencies of the fine structure components, and the stationary states will be those orbits whose parameters are given by expression (25). If we now consider the effect of external forces upon the fine structure components of the hydrogen lines it is necessary to keep in mind that this fixation of the stationary states only applies to the unperturbed hydrogen atom, and that, as mentioned, the orbits in these states are in general already strongly influenced by the presence of external forces, which are small compared with those with which we are concerned in experiments on the Stark and Zeeman effects. In general the presence of such forces will lead to a great complexity of perturbations, and the atom will no longer possess a group of sharply defined stationary states. The fine structure components of a given hydrogen line will therefore become diffuse and merged together. There are, however, several important cases where this does not happen on account of the simple character of the perturbations. The simplest example is a hydrogen atom perturbed by a central force acting from the nucleus. In this case it is evident that the motion of the system will retain its centrally symmetrical character, and that the perturbed motion will differ from the unperturbed motion only in that the frequency of rotation of the major axis will be different for different values of this axis and of the parameter. This point is of importance in the theory of the spectra of elements of higher atomic number, since, as we shall see, the effect of the forces originating from the inner electrons may to a first approximation be compared with that of a perturbing central field. We cannot therefore expect these spectra to exhibit a separate effect due to the variation of the mass of the electron of the same kind as that found in the case of the hydrogen lines. This variation will not give rise to a splitting up into separate components but only to small displacements in the position of the various lines.
We obtain still another simple example in which the hydrogen atom possesses sharp stationary states, although the change of mass of the electron is considered, if we take an atom subject to a homogeneous magnetic field. The effect of such a field will consist in the superposition of a rotation of the entire system about an axis through the nucleus and parallel with the magnetic force. It follows immediately from this result according to the principle of correspondence that each fine structure component must be expected to split up into a normal Zeeman effect (Lorentz triplet). The problem may also be solved by means of the theory of conditionally periodic systems, since the equations of motion in the presence of a magnetic field, even when the change in the mass is considered, will allow of a separation of the variables using polar coordinates in space. This has been pointed out by Sommerfeld and Debye.
A more complicated case arises when the atom is exposed to a homogeneous electric field which is not so strong that the effect due to the change in the mass may be neglected. In this case there is no system of coordinates by which the equations of motion can be solved by separation of the variables, and the problem, therefore, cannot be treated by the theory of the stationary states of conditionally periodic systems. A closer investigation of the perturbations, however, shows them to be of such a character that the motion of the electrons may be decomposed into a number of separate harmonic components. These fall into two groups for which the direction of oscillation is either parallel with or perpendicular to the field. According to the principle of correspondence, therefore, we must expect that also in this case in the presence of the field each hydrogen line will consist of a number of sharp, polarized components. In fact by means of the principles I have described, it is possible to give a unique fixation of the stationary states. The problem of the effect of a homogeneous electric field upon the fine structure components of the hydrogen lines has been treated in detail from this point of view by Kramers in a paper which will soon be published. In this paper it will be shown how it appears possible to predict in detail the manner in which the fine structure of the hydrogen lines gradually changes into the ordinary Stark effect as the electric intensity increases.
Theory of series spectra. Let us now turn our attention once more to the problem of the series spectra of elements of higher atomic number. The general appearance of the Rydberg constant in these spectra is to be explained by assuming that the atom is neutral and that one electron revolves in an orbit the dimensions of which are large in comparison with the distance of the inner electrons from the nucleus. In a certain sense, therefore, the motion of the outer electron may be compared with the motion of the electron of the hydrogen atom perturbed by external forces, and the appearance of the various series in the spectra of the other elements is from this point of view to be regarded as analogous to the splitting up of the hydrogen lines into components on account of such forces.
In his theory of the structure of series spectra of the type exhibited by the alkali metals, Sommerfeld has made the assumption that the orbit of the outer electron to a first approximation possesses the same character as that produced by a simple perturbing central field whose intensity diminishes rapidly with increasing distance from the nucleus. He fixed the motion of the external electron by means of his general theory for the fixation of the stationary states of a central motion. The application of this method depends on the possibility of separating the variables in the equations of motion. In this manner Sommerfeld was able to calculate a number of energy values which can be arranged in rows just like the empirical spectral terms shown in the diagram of the sodium spectrum ([p. 30]). The states grouped together by Sommerfeld in the separate rows are exactly those which were characterized by one and the same value of