The results he obtained as regards the fine structure of the hydrogen lines agree with observation inside the limit of experimental error. Although Sommerfeld’s methods have also been very fruitful when applied to the spectra of other elements, they were still of a purely formal and rather arbitrary nature; it is, therefore, of great importance that the Leiden professor, Ehrenfest, and Bohr succeeded later in handling the problem from a more fundamental point of view, Bohr making use of the correspondence principle previously mentioned. It should be said here, by way of suggestion, that Bohr used the fact that the motions of the electrons are not simple periodic but “multiple periodic.” We see this most simply if we think of the revolution of the electrons in the elliptical orbit as representing one period, and the rotation for the major axis of the ellipse as representing a second period. [Illustration: FIG. 27.—The model of the hydrogen atom with stationary orbits corresponding to principal quantum numbers and auxiliary quantum numbers.] Fig. 27 shows a number of the possible stationary orbits in the hydrogen atom according to Sommerfeld’s theory; for the sake of simplicity the orbits are drawn as completely closed ellipses. If we examine, for instance, the orbits with principal quantum number 4, we have here three more or less elongated ellipses, 4₁, 4₂, 4₃, and the circle 4₄; in all of them the major axis has the same length, and the length of the major axis is to that of the minor axis as the principal quantum number is to the auxiliary quantum number (for the circle 4 : 4 = 1). On the whole, to a principal quantum number _n_ there correspond the auxiliary quantum numbers 1, 2 ... _n_, and the orbit for which the auxiliary quantum number equals the principal quantum number is a circle. We see that in the more complicated hydrogen atom model there is possibility for a much greater number of different transitions than in the simple model (Fig. 25, p. 119). Some of the transitions are indicated by arrows. Since the energy content of the atom is almost the same for orbits with the same principal quantum number and different auxiliary quantum numbers, three transitions like 3₃-2₂, 3₂-2₁ and 3₁-2₂ will give about the same frequency, and therefore spectral lines which lie very close together. In a transition like 4₄-4₃ the emitted energy quantum _h_ν, and also ν, will be so extremely small that the corresponding line will be too far out in the infra-red for any possibility of observing it.