It is shown that the scattered field and the field inside the body can be expressed formally as power series in the above ratio, the calculation of successive terms in the series requiring the solution of standard problems in potential theory, together with the evaluation of certain potential integrals, so that the process can be carried as far as desired if Laplace's equation can be solved in the coordinate system appropriate for the body. The convergence of the series for the scattered field becomes progressively worse as we recede from the body, but an alternative expression for the field, also proceeding in powers of the same parameter, gives a representation which is valid everywhere except in the immediate neighborhood of the body (in particular, in the wave zone) and which does not suffer from this defect. The case of a perfect conductor, or diffraction through a hole in a perfectly conducting screen, can be treated as particular cases of the general theory. The paper can be regarded as an extension of Rayleigh's work, which confines itself to the first terms in the series.