Stability of Longitudinal Oscillations in a Uniform Magnetized Plasma with Anisotropic Velocity Distribution

We consider the longitudinal oscillations of a uniform magnetized plasma in which the particles are allowed to have anisotropic distributions in velocity space, corresponding to the two temperatures $T_{\perp }$ and $T_{\mathrm{II}}$ perpendicular and parallel to the magnetic field. It is shown that no instabilities can exist unless $T_{\perp }>T_{\mathrm{II}}$ for at least one of the species of plasma particles. An additional theorem, of use in surveying for possible unstable roots to the dispersion equation when $T_{\perp }>T_{\mathrm{II}}$, is also proved.