The propagation of radio waves of very low frequency to great distances is conveniently treated by regarding the space between the earth and the ionosphere as a wave-guide. Several authors have found that the least attenuated modes are profoundly affected by the earth's curvature. This effect is investigated for several models of the ionosphere. It is found, in particular, that for frequencies greater than about 30 ke/s some modes are possible for which the energy is concentrated in a region near the base of the ionosphere, and the field strength near the ground is small. It is useful to think of such modes as being composed of waves repeatedly reflected at the inside spherical surface of the ionosphere, the rays being chords of this sphere. By analogy with sound waves these modes are called `whispering gallery modes'. The theory uses wave admittance and reflexion coefficient variables because these satisfy differential equations which are convenient for integration using a digital computer. The curvature of the earth is allowed for by using the method of the modified refractive index, but the earth's magnetic field is neglected. Formulae for the mode condition and the excitation of the various modes by a transmitter are given and discussed. A new way of dealing with an ionosphere having a continuous electron density profile is presented. The results of some numerical calculations are given both for a sharply bounded homogeneous ionosphere and for an exponential profile of electron density.