This not only constitutes excellent evidence for the orbit theory, but it seems to be irreconcilable with a ring-electron theory once favored by some authors, since it requires the mass of the electron to be concentrated at a point.
The next amazing success of the orbit theory came when Sommerfeld[159] showed that the “quantum” principle underlying the Bohr theory ought to demand two different hydrogen orbits corresponding to the second quantum state—second orbit from the nucleus—one a circle and one an ellipse. And by applying the relativity theory to the change in mass of the electron with its change in speed as it moves through the different portions (perihelion and aphelion) of its orbit, he showed that the circular and elliptical orbits should have slightly different energies, and consequently that both the hydrogen and the helium lines corresponding to the second quantum state should be close doublets.
Now not only is this found to be the fact, but the measured separation of these two doublet lines agrees precisely with the predicted value, so that this again constitutes extraordinary evidence for the validity of the orbit-conceptions underlying the computation.
In [Fig. 27] the two orbits which are here in question are those which are labeled
and
; the large numeral denoting the total quantum number, and the subscript the auxiliary, or azimuthal, quantum number which determines the ellipticity of the orbit. The figure is introduced to show the types of stationary orbits which the extended Bohr theory permits. For total quantum number 1 there is but one possible orbit, a circle. For total quantum numbers 2, 3, 4, etc., there are 2, 3, 4, etc., possible orbits, respectively. The ratio of the auxiliary to the total quantum number gives the ratio of the minor and major axes of the ellipse. The fourth quantum state, for example, has four orbits,
,