When the magnetosphere is magnetically open, the magnetopause must cut across open field lines and is usually identified with a rotational discontinuity or finite-amplitude Alfv¿n wave. The merging of magnetospheric and interplanetary magnetic fields in the nose region of the magnetosphere generates two pairs of such Alfv¿n waves the two interior ones propagate down to the ionosphere, and the two exterior ones propagate into the incoming magnetosheath plasma and thus can be identified with the magnetopause in the dayside. However, continuing this magnetopause into the nightside magnetotail region would require reversing the direction of magnetic-field rotation at some point on the propagating wave, which is not possible. To resolve this problem, I propose that the magnetopause on the nightside is not a continuation of that on the dayside but is a distinct surface formed by a separate Alfv¿n wave generated near the down-stream edge of the dayside polar cusp. This multiple-branch model also explains the boundary of the exterior cusp (identified with the continuation of the dayside magnetopause) and accounts for the non-stagnant flow within the exterior cusp (the flow simply proceeds across the nightside magnetopause into the plasma mantle region). Where the two branches are adjacent to the magnetotail, their spatial separation can be estimated as typically no more than about 2RE, not likely to have been noticed unless specifically looked for. A possible mechanism for generating the nightside Alfv¿n wave is the expansion of plasma within the polar cusp as the flux tubes are advected into the rapidly increased volume of the magnetotail. ¿American Geophysical Union 1995