Since the initial work of Brillouin, interest in long, dense electrom beams with a uniform velocity profile has increased steadily. Notable contributions have been made by Samuel and Harris. In this paper, the general member of this class of beams is derived. Various special cases become the equilibrium systems of current interest. In addition to these known systems, the analysis leads to new possibilities. One of these is of particular interest in that it is the first of this group of beams in which the axial velocity is important to the balance of forces on the electrons. The new beam has properties ideal forM‐type interaction and should be of considerable future importance.