When we try to apply assumptions, analogous with C and D, to systems containing more than one electron, we meet with difficulties, since in this case the application of ordinary mechanics in general does not lead to periodic orbits. An exception to this, however, occurs if the electrons are arranged in rings and rotate in circular orbits, and from simple considerations of analogy the following assumption was proposed (see I. p. 24).
E. In any atomic or molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relative to each other, and the electrons move in circular orbits, the angular momentum of each electron round the centre of its orbit will be equal to
in the “normal” state of the system, i.e. the state in which the total energy is a minimum.
It was shown that in a number of different cases this assumption led to results in approximate agreement with experimental facts. In general, no stable configuration in which the electrons rotate in circular orbits can exist if the problem of stability is discussed on ordinary mechanics. This is no objection, however, since it is assumed already that the mechanics do not hold for the transition between two stationary states. Simple considerations led to the following condition of stability.
F. A configuration satisfying the condition E is stable if the total energy of the system is less than in any neighbouring configuration satisfying the same condition of angular momentum of the electrons.
As already mentioned, the foundation for the hypothesis E was sought in analogy with the simple system consisting of one electron and one nucleus. Additional support, however, was obtained from a closer consideration of the formation of the systems. It was shown how simple processes could be imagined by which the confluence of different rings of electrons could be effected without any change in the angular momentum of the electrons, if the angular momentum of each electron before the process was the same. Such considerations led to a theory of formation of molecules.
It must be emphasized that only in the case of circular orbits has the angular momentum any connexion with the principles of the Quantum theory. If, therefore, the application of ordinary mechanics to the stationary states of the system does not lead to strictly circular orbits, the assumption E cannot be applied. This case occurs if we consider configurations in which the electrons are arranged in different rings which do not rotate with the same frequency. Such configurations, however, are apparently necessary in order to explain many characteristic properties of the atoms. In my previous papers an attempt was made in certain cases to overcome this difficulty by assuming, that if a very small alteration of the forces would make circular orbits possible on ordinary mechanics, the configuration and energy of the actual system would only differ very little from that calculated for the altered system. It will be seen that this assumption is most intimately connected with the hypothesis F on the stability of the configurations. Such considerations were used to explain the general appearance of the Rydberg constant in the spectra of the elements, and were also applied in discussing possible configurations of the electrons in the atoms suggested by the observed chemical properties. These calculations have been criticised by Nicholson[8], who has attempted to show that the configurations chosen for the electrons in the atoms are inconsistent with the main principles of the theory, and has also attempted to prove the impossibility of accounting for other spectra by help of assumptions similar to those used in the interpretation of the hydrogen spectrum.
Although I am quite ready to admit that these points involve great and unsolved difficulties, I am unable to agree with Nicholson’s conclusions. In the first place, his calculations rest upon a particular application to non-circular orbits of the principle of constancy of angular momentum for each electron, which it does not seem possible to justify either on the Quantum theory or on the ordinary mechanics, and which has no direct connexion with the assumptions used in my papers. It has not been proved that the configurations proposed are inconsistent with the assumption C. But even if it were possible to prove that the unrestricted use of ordinary mechanics to the stationary states is inconsistent with the configurations of the electrons, apparently necessary to explain the observed properties of the elements, this would not constitute a serious objection to the deductions in my papers. It must be remarked that all the applications of ordinary mechanics are essentially connected with the assumption of periodic orbits. As far as the applications are concerned, the first part of the assumption C might just as well have been given the following more cautious form:—
“The relation between the frequency and energy of the particles in the stationary states can be determined by means of the ordinary laws of mechanics if these laws lead to periodic orbits.”