The results of a theoretical study of the use of electrostatic fields to suppress the plasma losses from magnetic cusps or mirrors are reported. Numerical estimates are given for the maximum plasma densities which can be contained. Assuming stability, the use of electrostatic fields, combined with controlled electron and ion beam injection, makes it possible t o generate cusps with a width of 4 ρe, where ρe is the electron Larmor radius. With such narrow cusps, densities which are sufficiently high for a fusion reactor can be contained electrostatically. High values of the electric stress tensor (E2/8π) are required near the positive electrodes but the magnitude of these values is shown to be appreciably less than the plasma pressure and to be within practical limits. When the electrostatic fields are combined with a non-adiabatic cusp magnetic field, a form of particle beam injection is possible which is orders of magnitude more efficient than existing methods. The main body of the plasma should have a high degree of stability; the only weak regions for stability are in the cusps, where the electrons have been separated from the ions. The conditions in the cusps are greatly different from any model which has been treated by stability theory; the finite dimensions and non-uniformity in both the longitudinal and transverse directions will have stabilizing effects and the resultant stability of the electron cusps is an open question.