When, now, the hydrogen nucleus is replaced by that of helium, which is four times as heavy, the common center of gravity is still closer to the nucleus, so that the helium-nucleus describes a much smaller circle than did that of hydrogen. This situation is responsible for a slight but accurately predictable difference in the energies of the two orbits, which should cause the spectral lines produced by electron-jumps to these two different orbits to be slightly displaced from one another. This predicted slight displacement between the hydrogen and helium lines is not only found experimentally, but the most refined and exact of recent measurements has shown that the observed displacement agrees with the predicted value to within a small fraction of 1 per cent. 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. [Pg 218] 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, [Pg 219] has four orbits, , , , , all of which have the same major axis, but minor axes which increase in the ratios 1, 2, 3, 4 up to equality, in the circle (), with the major axis. It is this multiplicity of orbits which predicts with beautiful accuracy the “fine-structure” of all of the lines due to atomic hydrogen and to helium.