. Taking the variation of the mass of the electron with velocity into account and neglecting terms of higher order of magnitude he obtained
where
is the velocity of light.
Corresponding to each of the energy values for the stationary states of the hydrogen atom given by the simple formula (5) we obtain
values differing only very little from one another, since the second term within the bracket is very small. With the aid of the general frequency relation (1) we therefore obtain a number of components with nearly coincident frequencies instead of each hydrogen line given by the simple formula (2). Sommerfeld has now shown that this calculation actually agrees with measurements of the fine structure. This agreement applies not only to the fine structure of the hydrogen lines which is very difficult to measure on account of the extreme proximity of the components, but it is also possible to account in detail for the fine structure of the helium lines given by formula (7) which has been very carefully investigated by Paschen. Sommerfeld in connection with this theory also pointed out that formula (11) could be applied to the X-ray spectra. Thus he showed that in the
and