[11]. The theoretical value for the ratio between the second factor in (3) for this spectrum and for the hydrogen spectrum is 1.000409; the value calculated from Fowler’s measurements is 1.000408[12]. Some of the lines under consideration have been observed earlier in star spectra, and have been ascribed to hydrogen not only on account of the close numerical relation with the lines of the Balmer series, but also on account of the fact that the lines observed, together with the lines of the Balmer series, constitutes a spectrum which shows a marked analogy with the spectra of the alkali metals. This analogy, however, has been completely disturbed by Fowler’s and Evans’ observations, that the two new series contain twice as many lines as is to be expected on this analogy. In addition, Evans has succeeded in obtaining the lines in such pure helium that no trace of the ordinary hydrogen lines could be observed[13]. The great difference between the conditions for the production of the Balmer series and the series under consideration is also brought out very strikingly by some recent experiments of Rau[14] on the minimum voltage necessary for the production of spectral lines. While about 13 volts was sufficient to excite the lines of the Balmer series, about 80 volts was found necessary to excite the other series. These values agree closely with the values calculated from the assumption E for the energies necessary to remove the electron from the hydrogen atom and to remove both electrons from the helium atom, viz. 13.6 and 81.3 volts respectively. It has recently been argued[15] that the lines are not so sharp as should be expected from the atomic weight of helium on Lord Rayleigh’s theory of the width of spectral lines. This might, however, be explained by the fact that the systems emitting the spectrum, in contrast to those emitting the hydrogen spectrum, are supposed to carry an excess positive charge, and therefore must be expected to acquire great velocities in the electric field in the discharge-tube.
In paper IV. an attempt was made on the basis of the present theory to explain the characteristic effect of an electric field on the hydrogen spectrum recently discovered by Stark. This author observed that if luminous hydrogen is placed in an intense electric field, each of the lines of the Balmer series is split up into a number of homogeneous components. These components are situated symmetrically with regard to the original lines, and their distance apart is proportional to the intensity of the external electric field. By spectroscopic observation in a direction perpendicular to the field, the components are linearly polarized, some parallel and some perpendicular to the field. Further experiments have shown that the phenomenon is even more complex than was at first expected. By applying greater dispersion, the number of components observed has been greatly increased, and the numbers as well as the intensities of the components are found to vary in a complex manner from line to line[16]. Although the present development of the theory does not allow us to account in detail for the observations, it seems that the considerations in paper IV. offer a simple interpretation of several characteristic features of the phenomenon.
The calculation can be made considerably simpler than in the former paper by an application of Hamilton’s principle. Consider a particle moving in a closed orbit in a stationary field. Let
be the frequency of revolution,
the mean value of the kinetic energy during the revolution, and
the mean value of the sum of the kinetic energy and the potential energy of the particle relative to the stationary field. We have then for a small arbitrary variation of the orbit
