A new representation is developed for that portion of the magnetospheric magnetic field which is produced by currents on the magnetopause or outside it. This field is represented by a scalar potential &ggr;, expanded in parabolic harmonics: it may be obtained, in first approximation, by regarding the solar wind as a perfect conductor which excludes all fields of magnetospheric sources (Chapman-Ferraro boundary conditions). For a magnetopause which is a paraboloid of rotation and which confines all field lines of magnetospheric sources, the fields due to magnetopause currents are derived for two sources, a dipole and a two-parameter model of the ring current, each arbitrarily inclined to the earth-sun line. Methods are also developed for accomodating changes in the size of the magnetospheric cavity and for using nonparaboloidal boundaries, open at the nightside. Neutral points and polar cusps are derived, and in agreement with observations it is found that a growth of the ring current shifts them equatorward. By extrapolation from these results, it is argued that the observation of such shifts during magnetic storms, together with Dst, may help resolve the relative importance of two processes proposed for magnetic storms.