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The radiation fields produced by a sinusoidal distribution of axial-electric field along a thin circumferential slot, cut in a perfectly conducting infinite cylinder which is covered by a concentric dielectric coating, are found by applying the method of Wait with two modifications. Initially, a structure consisting of a finite coated cylinder containing the slot and exciting a radial waveguide is considered. The fields in this waveguide are expressed in terms of derivatives of two axial Hertz vectors, and for a finite radial wall spacing are seen to consist of a double Fourier series. The radial walls are then allowed to become infinitely spaced; in this process, the Fourier-series representation for the axial dependence of the fields becomes a Fourier integral. The radiation fields are then found by asymptotically evaluating the Fourier integral by the method of stationary phase rather than by the saddle-point method. Expressions for the radiation fields are then found, but are given explicitly for the equatorial plane only. Calculated radiation patterns in this plane, for a coating having a fixed dielectric constant with the thickness as a parameter, and for a fixed thickness with the dielectric constant as a parameter, are given for the case of a cylinder of size a=3. Generalizations based on these calculations are suggested. The case of a specific plasma coating is briefly considered, and an approximate solution, readily obtained, which gives the same azimuthal form for the equatorial patterns, is also noted.