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The radiation properties are investigated for an axial slot antenna on a conducting elliptic cylinder with a homogeneous dielectric coating. In the dielectric coating and in the exterior free-space region, the field is expanded in elliptic waves via the Mathieu functions. The Mathieu angular functions are employed as basis and testing functions to enforce the boundary conditions via Galerkin's method at the interface between the dielectric and the free-space regions. Numerical results are presented in graphical form for the transverse electric (TE) and transverse magnetic (TM) polarizations to illustrate the far-field radiation patterns, the gain versus coating thickness, and the aperture conductance versus coating thickness.