A study of the two-dimensional incompressible flow of the solar wind past an open terrestrial magnetosphere is presented. The plasma is described by the magnetohydrodynamic (MHD) equations including non-ideal effects such as resistivity and viscosity. The theoretical analysis is based on an ordinary perturbation technique with expansion in orders of large Reynolds and Lundqvist numbers. Magnetic reconnection is assumed to occur at an arbitrary point along a region of the magnetopause, stretching from the sub-solar point to the north. From the analysis we find analytical solutions describing the transition layers north and south of a reconnection point. It is shown that the transition layer north of the reconnection point is thinner than the layer on the south side. It is also shown that in order for steady reconnection to occur, the velocity at the reconnection point must be less than the Alfvén velocity. Using the method of matched asymptotic expansions, we match the transition layer solution with a potential flow solution describing the flow and magnetic field in the exterior magnetosheath.