The problem of ground-wave propagation over an inhomogeneous smooth earth is discussed in terms of the known solution for a homogeneous earth. The inhomogeneity refers only to changes in the earth constants from place to place, and the problem is idealized by assuming a wave radiated from a vertical dipole over a series of homogeneous annular sections. After a statement of some of the conditions the solution would be expected to obey, including the essential one of reciprocity, some fundamental results of the theory for a homogeneous earth are described in a form directly useful to the argument. The solution is first given for the short-wave limit, where it is complete except in the neighbourhood of a boundary. By an approximate consideration of the energy flow at different heights above the ground, the solution is extended to the case of intermediate wavelengths where the first and last boundaries are in the diffraction region of the transmitting and receiving points respectively. It is then shown that a well-known empirical method yields the same solution when it is made reciprocal by taking the geometric mean of the value it gives and the value that would be obtained with the transmitter and receiver interchanged. This method is formally used to obtain a tentative solution for the effect of the disturbance function in the neighbourhood of a boundary. It leads to the striking suggestion that on passing from a section of one value of conductivity to another of a higher one, there is a recovery in field-strength before the attenuation of the wave becomes characteristic of the new section. On crossing the boundary in the other direction, there is a correspondingly increased drop in field strength before the attenuation takes its new characteristic type. Owing to the lack of sufficiently controlled conditions, most of the existing experimental results are inconclusive with regard to these features at a boundary, but some evidence is given in support of them. Stress is laid on the need for further experiments specifically designed to study the field near a land-sea boundary. The paper deals briefly with the practical application of the method, and gives a specimen field-strength/distance curve for a route consisting of several land and sea sections. It concludes by pointing out that further research is needed, especially with regard to the phase relationships, as the argument has dealt only with field-strength values.