The behaviour of very low frequency (V.L.F.) radio waves propagating in an ionlzed medium, such as the ionosphere and exosphere, may exhibit markedly different characteristics from those of frequencies such as are used for communication in the short-wave bands. The anisotropy introduced into the medium by the presence of the earth's magnetic field has a significant effect, while at frequencies below about 5KHz the presence of heavy positive ions may further affect propagation. In particular it is found that, under suitable conditions, V.L.F. signals may propagate along paths closely following the lines of force of the earth's magnetic field. The study of such signals has a practical interest in that, from consideration of their propagation times and dispersion properties, they can provide an inexpensive method of determining electron densities at well beyond the range of satellite investigation. For this reason, if for no other, theoretical investigation of the paths traced in a model exosphere may indicate how closely the model exosphere approximates the true one. A powerful method of tracing out these paths is by the use of "ray-tracing", provided the properties of the medium vary slowly in space, a condition not always fulfilled in practice. It has been the subject of this thesis to carry out ray-tracing studies, in particular noting the effects of the positive ions in the exosphere; effects which have not received much notice before as they were thought to be unimportant. Intro., p. 1-2.