An attempt is made in the paper to provide a satisfactory theoretical basis for a future discussion of the experimental data on the propagation of very long radio waves (18,800 m.) given by Best, Ratcliffe & Wilkes, and Budden, Ratcliffe & Wilkes. The reflexion of very long plane waves incident obliquely on a horizontally stratified ionized medium with a vertical magnetic field is first considered in general terms, and it is shown that the medium can be divided into a transition region and a reflecting region. If the ionization in the reflecting region increases linearly with height it is shown that propagation is governed by the following equations: ∂
2
L
/ ∂ζ
2
+ (α + ζ)
L
+ β
M
= 0, ∂
2
M
/ ∂ζ
2
+ (α - ζ)
M
+ β
L
= 0, where α and β are constants depending on the angle of incidence. Under the conditions of the experiments β is small, and a solution, in terms of contour integrals, valid in this case is obtained.