The 3-dimensional potential due to a charged circle can be derived in various ways which lead to the same expression in terms of the toroidal coordinate σ through Legendre functions of s (= cosh σ). Applying the same methods to the magnetic potential of a circular current we find various results: (a) Legendre's equation yields no solution (§4·1). (b) The appropriate transformation from cylindrical to toroidal coordinates gives a Fourier series (equation (48)) the coefficients of which, though not Legendre functions, are related to them by the identity (§4·2) By this method we also derive expressions in elliptic integrals (equations (56), (58)). (c) Direct integration of the solid angle gives an alternative expression in elliptic integrals (equation (59)).