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Infinite series involving associated Legendre functions of general degree and order are used to describe the scalar potential in the vicinity of a cubic corner. The method of analysis closely follows methods used to describe the magnetic field in the gap of a ring head for magnetic recording and the degree of the Legendre functions can be approximated when the determinant of the derived matrix vanishes. We solve with both Dirichlet and Neumann boundary conditions, but study symmetric and antisymmetric solutions separately. Other geometries considered are the two-dimensional straight edge and the quarter-plane. We also discuss the degeneracy of higher order eigenvalues.