Bohr's success in accounting for the frequencies observed in the spectral series of hydrogen was rightly regarded as a triumph, despite the fact that it violated the classical laws of mechanics and electromagnetism. It could not account, however, for the spectra of helium and heavier elements. The Bohr model was the simplest possible model for the dynamics of a single electron in the electrostatic potential of a positively charged point nucleus, in that it involved only quantised circular orbits defined by a single quantum number n, what became known as the principal quantum number. At the 1911 Solvay Conference, before Bohr's announcement of his model for the hydrogen atom, Poincaré had raised the issue of how the quantisation conditions could be extended to systems of more than one degree of freedom. The problem was attacked by both Planck and Sommerfeld. Their approaches ended up being essentially the same, although expressed in somewhat different language. We will follow Sommerfeld's approach.

In 1891 Michelson had shown that the Hɑ and Hß lines of the Balmer series displayed very narrow splittings (Michelson, 1891, 1892). Although incompatible with Bohr's theory, the problem was set aside in the face of the other remarkable successes of the theory. Sommerfeld suspected that the explanation lay in the fact that Bohr's quantisation condition involved only a single degree of freedom. In his papers of 1915 and 1916, he extended the quantisation of the orbits of the electron to more than one degree of freedom and accounted for the splitting of the lines of the Balmer series once a special relativistic treatment of the model was adopted (Sommerfeld, 1915a,b, 1916a).