It is shown that the relativistic wave equation for electrons in a uniform magnetic field leads to the same wave function as that already deduced by Page from the non-relativistic equation. As in the latter case the motion at right angles to the field is quantized.An expression is found for the current density from the relativistic wave equation. The relativistic expression differs from the non-relativistic only by a constant factor which does not affect the calculation of the mean radii of curvature of the electron current.Hence, for the relativistic case, as for the non-relativistic, the mean radius of curvature is less than that expected on the classical theory. It follows that the classical relativistic relation between $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ and the mean radius of curvature upon deflection gives a value of $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ which is too large.
The validity of de Broglie's equation is checked experimentally for electrons of 24 to 64 electron-kilovolts energy. The experiment consists essentially of deflecting a beam of cathode rays by an electrostatic field and, keeping the deflection constant, determining the relation between the intensity of the deflecting field,
V, and the de Broglie wave-length, $\ensuremath{\lambda}$, of the deflected electrons. On the assumption that the motion of the electrons is in accordance with the laws of relativistic mechanics, and that the de Broglie wave-length is inversely proportional to the momentum, it is shown theoretically that the expression ${\ensuremath{\lambda}}^{2}V{(1+\frac{{h}^{2}}{{{m}_{0}}^{2}{c}^{2}{\ensuremath{\lambda}}^{2}})}^{\frac{1}{2}}$ should be independent of $\ensuremath{\lambda}$. Within the limits of experimental error this expression is found to be independent of $\ensuremath{\lambda}$, thereby demonstrating that the de Broglie equation, $\ensuremath{\lambda}=\frac{h}{\mathrm{mv}}$, combined with the relativity expression for
m, is adequate to describe the facts.
The wave equation for a stream of electrons passing through a uniform magnetic field is solved and the characteristic functions discussed. The motion at right angles to the field is found to be completely quantized.The current density is investigated, and the mean square and mean fourth power radii of curvature found for each quantum state. The mean square radius of curvature is found to be the same for all quantum states of the same energy and identical with that predicted by the classical theory.The mean radius of curvature is determined by interpolation and is found to be less than that predicted by the classical theory. Therefore values of $\frac{e}{\ensuremath{\mu}}$ calculated by means of the classical formula from measured values of the mean radius of curvature should be too large. The error is estimated and found to be of the same general magnitude as the discrepancy between the results of deflection experiments and those of spectroscopic measurements.