The wave-fields inside a horizontally stratified ionosphere produced by a radio wave incident at any angle from below are calculated by integrating the governing differential equations. An improved method of obtaining two independent solutions of these equations is described in some detail. For the first solution, the polarization of the wave incident on to the ionosphere from below is chosen so that no energy is propagated to great heights (i.e. no energy in the `whistler mode'); the second solution has the incident polarization chosen to maximize this energy. In each case, a complex reflexion coefficient is derived from the relative amplitude and phases of the upgoing and downgoing waves in free space below the ionosphere, and the polarizations of the downgoing wave are also studied. The method which is usable on a small computer, is applicable for any angles of incidence, azimuth and dip, and for any height variation of the model ionosphere.