This thesis deals with the instabilities which can exist in infinite, uniform, collisionless plasmas having non-Maxwellian distributions of particle velocities. The instabilities are treated in terms of exponentially growing linearized plane waves in the plasma. The existence and properties of these waves can be determined from certain "dispersion relations", or equations relating the frequency of the waves to their wavelength. These dispersion relations are exhibited for all classes of linearized plane waves, and a formal solution of the stability problem is given. A new analogy to electrostatic potential theory, and a classical method, the Nyquist diagram, are used separately and in combination to reduce the formal solution to practicable techniques in several important restricted cases. For example, solutions are obtained to the problems of stability with respect to longitudinal waves in the absence of a D.C. magnetic field, and of stability with respect to longitudinal and to transverse waves propagating along a D.C. magnetic field.
In the course of the analysis it is found that a deficit of particles at a certain velocity tends to produce growing longitudinal oscillations of that phase velocity, while an excess tends to produce growing transverse waves. An arbitrarily small deficit or excess can still produce instability if it involves abrupt enough variations of the particle densities in velocity space.
Two examples are presented which are important in astrophysics for understanding the formation of shock fronts, and one example is given which may be of value in explaining D.C. plasma resistivity at high temperatures, when binary collisions are negligible. Certain instabilities in counterstreaming plasmas that have been used in the literature to determine hydromagnetic shock thicknesses, and to explain the presence of abnormally high energy electrons in the outer Van Allen belt or belts, are found to be absent unless the initial temperatures of the plasmas are extremely low.
