A theory of electrodeless electric propulsion systems (EEPS) based on the use of the solenoid magnetic field and the rotating electromagnetic field produced by antennas is developed, which includes a study of the plasma acceleration by the Radio Frequency (RF) field and the concomitant thrust. It was assumed that the frequency of the RF field exceeds the lower hybrid frequency but is much less than the electron gyrofrequency. Relations for the thrust are obtained and analyzed. It is shown that thrust gain is significant only when the RF-induced drift velocity well exceeds the fluid velocity of the injected plasma. It is revealed that the curvature of the magnetic field lines and the plasma acceleration in the region outside the solenoid are the factors which can considerably increase the thrust. On the other hand, it is found that the axial inhomogeneity of the plasma and some other factors are unfavorable for the thrust. The obtained results can be used for the optimization of particular experiments aimed to create a new thruster for long-time space missions.