Sinusoidal electron charge density fluctuations with propagation vector k are considered for a fully ionized gas in complete thermodynamic equilibrium in a constant magnetic field. Let $\alpha$ and $\epsilon$ be the ratio of the Debye length and of an electron gyroradius, respectively, to the wavelength $k^{-1\mathrm{}}$. A general formula is derived for the frequency spectrum of these fluctuations for arbitrary values of $\alpha$, $\epsilon$, and of the angle ($\phi -\frac{1}{2}\pi$) between k and the magnetic field. The dispersion relation implied by this expression has been obtained previously by Gross and by Bernstein, but the method of derivation is different. A very small electron-ion mass ratio $\frac{m}{M}$ is assumed.

For large values of $\epsilon$ and $\alpha$ most of the intensity occurs at small frequencies: If $sin\phi \gg {\left(\frac{m}{M}\right)}^{\frac{1}{2}}\epsilon$, the main spectrum is continuous as in the absence of a magnetic field; if ${\left(\frac{m}{M}\right)}^{\frac{1}{2}}\ll sin\phi \ll {\left(\frac{m}{M}\right)}^{\frac{1}{2}}\epsilon$, it consists of lines with spacing about the ion gyrofrequency; if $sin\phi \ll {\left(\frac{m}{M}\right)}^{\frac{1}{2}}$, it consists mainly of a line at zero frequency. Weaker spectral lines are obtained which correspond to plasma oscillations, the existence of "frequency gaps" is confirmed for small angles $\phi$, and the intensities of the various components are evaluated. For small $\phi$, another spectral line is obtained at a "resonance" frequency intermediate between the electron and ion gyrofrequency.

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