This work is general although it has been done with the ionosphere in mind. Such a nonuniform plasma presents exceedingly difficult conditions to a complete solution. That is, the index of refraction is a function of the operating frequency, hence such medium is dispersive and the various spectral components of a radiated waveform travel at different velocities; the medium is inhomogeneous and varies three dimensionally, particularly in regard to permittivity, index of refraction, and velocity of spectral components of waveform; the medium is anisotropic and hence the permittivity and related quantities are tensors. This means that the medium is birefringent and that for wave frequencies close to the gyromagnetic frequency the ray is bidirectional, consisting of the ordinary and the extraordinary wave. The general propagation problem is treated, including absorption. Then there is an outline of a treatment of exoionospheric propagation as a boundary-value problem and its programmation for the digital computer. Then the Hamiltonian optics ray-tracing theory is treated. This effectively reduces the pertinent-ray tracing problem to the solution of six equations which describe the ray position and associated wave normal (by adding another two equations the absorption and equivalent path are also obtained). In this very general treatment, the specific problems of interest fall out when the proper conditions are inserted. These may include the effects of irregularities in the electron density distribution, discontinuities as in going from light to darkness, holes, magnetic field variations, etc., which are evaluated simply by introducing these phenomena into the electron distribution model itself and in the expression involving the magnetic field contained in the Appleton-Hartree formula. The whistler modes of propagation may be given simply by using frequencies less than the gyromagnetic.
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