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The stability of a finite thickness, laminar cylindrical shell of electrons rotating azimuthally and enclosed in a coaxial waveguide is considered. The equilibrium rotation of the electrons is supported either by a radial electric field, an axial magnetic field, or a combination of both. The stability problem is formulated exactly as an eigenvalue problem, including all relativistic and electromagnetic effects as well as all effects of self and applied equilibrium fields. An approximate dispersion relation, valid for thin beams, is obtained analytically and the classical results for the ‘‘longitudinal’’ modes, i.e., the negative mass, cyclotron maser, and diocotron instabilities and for the ‘‘transverse’’ mode are recovered in appropriate limits. The dispersion relation is relatively simple and is valid for arbitrary values of the equilibrium electric and magnetic fields and for arbitrary beam energy. It therefore provides a ready comparison of the small signal properties of such devices as the Astron, gyrotron, orbitron, heliotron and the various cross field devices. It may also be of interest in accelerator and space physics applications. Some heretofore unnoticed effects on beam stability of equilibrium fields are reported; one such effect in particular leads to formulation of a simple, effective method either to maximize or eliminate altogether the longitudinal mode growth. Results from the dispersion relation compare favorably to results obtained from a numerical solution of the eigenvalue problem.