It is shown that resonant Alfv¿n waves on dipole magnetic field geometry and plasma distributions suitable for auroral field lines can be conveniently treated in the theory of Mathieu functions. Resurgent interest in invoking large-scale Alfv¿n waves to structure some elements of auroral electrodynamics calls for interpretation of measured perpendicular electric and magnetic disturbance fields in terms of Alfv¿n waves. The ability to express the resonant eigenmodes in closed form in terms of Mathieu functions allows for convenient tests of the Alfv¿n wave structuring hypothesis. Implications for current vector electric and magnetic disturbance measurements are discussed. ¿American Geophysical Union 1987