Analytic solutions are presented for magnetospheric convection which include the effects of energetic ions and a radial gradient of the plasma content of flux tubes. The latter assumption leads to results that differ from previous studies. These include (1) distributed, rather than sheet, Birkeland currents, (2) simple expressions for the oval, its thickness, its flow, its currents, and (3) Harang discontinuities and their properties, including location, length, and the angle they make with the oval. The effect on the above of differing nightside boundary conditions is presented. Some numerical solutions of cases with a weak azimuthal gradient of plasma content were also studied. The best semblance to observational models is for a maximum of the nightside earthward convection in the premidnight quadrant and a weak component of the gradient of plasma content along the oval from day to night. The calculated plasma energy distribution is reasonably consistent with the observed temperature of the plasma sheet in the energy range of importance, suggesting that the approach used to introduce self- consistency is successful.