A theory of the small-amplitude oscillations of an ionized gas in a static magnetic field is developed, including the effects of temperature motions. The Boltzmann equation is solved for this problem, and exact expressions are obtained for the distribution function and dispersion relation. A general feature of the dispersion relation is the existence of gaps in the spectrum at frequencies which are approximately multiples of ${\omega }_c=\frac{\mathrm{eH}}{\mathrm{mc}}$. The magnitude of the gap depends on the temperature of the gas, being proportional to it for long wavelengths. This leads to the prediction of selective reflection of waves impinging on a plasma with frequency in the forbidden range.

For $\mathrm{ck}\gg {\omega }_p$, ${\omega }_c$ the waves split into approximately longitudinal plasma waves and transverse waves. Detailed analysis is made of the plasma waves for ${\omega }_c$ small and ${\omega }_c$ large. At long wavelengths the frequency is ${\omega }^2\simeq \omega _p^{}{}_{}{}^2+\omega _c^{}{}_{}{}^2+\beta \left(\frac{\kappa T}{m}\right)k^2$, where $\beta$ depends on ${\omega }_p$ and ${\omega }_c$. For waves near the Debye length the waves are heavily damped.

Two simplified treatments of plasma oscillations based on transport equations are compared with the above treatment. Expressions of the form ${\omega }^2\simeq \omega _p^{}{}_{}{}^2+\omega _c^{}{}_{}{}^2+\beta \left(\frac{\kappa T}{m}\right)k^2$ are obtained where the factor $\beta$ is independent of ${\omega }_c$ and ${\omega }_p$. In addition, the transport treatments fail to predict the heavy damping near the Debye length and the existence of gaps in the frequency spectrum.

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