The flux tube and electric current formulations of magnetospheric convection are developed in parallel. The magnetosphere is modeled as consisting of three types of flux tubes; polar cap, auroral oval (plasma-rich flux tubes), and lower latitude. In each region the velocity can be expressed as the gradient of a potential which obeys the LaPlace equation. Boundary conditions between the three regions are that normal velocity and magnetic plus plasma pressure be continuous. The oval is much longer than it is wide (latitudinal extent) and approximately east-west oriented. This allows a very useful approximation, the narrow-channel approximation to be used which is readily solvable and can be used to obtain flows in all three regions. The validity of this approixmation is enhanced by the high conductivity of the oval. Application of this approximation to the oval results in the velocity along the oval being the gradient of a potential which is a function of the location or the oval boundaries. Thus flow in the oval can be solved from its boundary location of vice versa. The resulting solution provides boundary conditions for the other regions. The approximation is applied to a number of problems. The no-flow solution consists of zero thickness for the oval, and the polar cap is circular, centered nightward of the pole. It is surmised that a minimum-energy criterion applies to the magnetosphere. That is, it contains the minimum amount of auroral oval flux permitted for a given cross-polar cap potential. This implies zero thickness for the oval at noon. Convecting magnetospheres are considered next. An isostasy principle holds in that a bulge of the oval into the polar cap must be balanced by a bulge to lower latitudes at the same longitude. The flow equations were applied to the observed Feldstein ovals. The results showed velocities up to 1 km/s, cross-polar cap potentials up to 110 kV, and intense inflow in a narrow region near midnight. These all vary systematically with the Q magnetic-activity index and local time. The application of the equations to particles with significant field gradient drift indicated three differences: (1) a mid- and low-latitude flow of several kilovolts occurs in the steady state, (2) a dawn-dusk asymmetry is produced in the thickness of the oval, and (3) the form and slope of the Harang discontinuity are predicted. The introduction of ionospheric conductivity variation effects results in the following conclusions: (1) average velocities along the oval are unaffected, (2) the presence of the westward (eastward) electrojet displaces the oval to higher (lower) latitudes by distances of several hundred kilometers, and (3) the electrojets close by both partial ring currents and magnetotail currents.