The study of very long-distance and around-the-world propagation of HF radio waves becomes more urgent in connection with the problems of long distance ground-based radio communications, communications with space crafts and satellites, satellite-to-satellite communications, around-the-world radar scanning, around-the-world sounding of the ionosphere, etc. At pre sent, these investigations have acquired particular interest because transmit ters which make it possible to intentionally modify the ionospheric properties by powerful radio waves have become available. In the case of radio wave propagation over comparatively small distances (about 3000 - 5000 km, one - two hops), in a first approximation, the iono sphere can be considered homogeneous in the horizontal direction. The radio wave propagation theory in a horizontally-homogeneous, i. e. , spherically symmetric ionosphere was developed with sufficient completeness as early as in the 1920-1940’s by Appleton, Ratcliffe, Beynon, Booker, Martyn, and others. This theory is presented in detail in the well-known monographs by Ginzburg (1967), Bremmer (1946), and Budden (1961). Based on this theory, detailed methods for the calculation of radio paths, determination of field amplitude, and interpretation of vertical and oblique ionograms have been developed. All these methods are well-known and widely used in practice, see monographs by Al'pert (1974), Shchukin (1940), and Davies (1969). An altogether different situation takes place in the case of very long-dis tance multihop and around-the-world propagation.