Symmetry properties in a two-stream electrostatic instability

A double stream consisting of immobile ions and mobile electrons which is partly drifting and partly stationary is considered. Certain symmetry properties of the dispersion relation of the longitudinal plasma wave having its wave vector parallel to the direction of the electron stream are discussed. These symmetry properties enable the results for a particular value of the relative number density of the streaming electrons to be deduced from the corresponding results for the same value of the relative number density for the stationary electrons.