A study is made of the vibration characteristics of a cylindrical shell with arbitrary boundary conditions and with several intermediate constraint positions between the ends. The solution is obtained using a Rayleigh-Ritz procedure in which the axial modal displacements are constructed from simple Fourier series expressions. Geometric boundary conditions that are not identically satisfied are enforced with Lagrange multipliers. Unwanted geometric boundary conditions, forced to be zero due to the nature of the assumed series, are released through the mechanism of Stokes' transformation. For the problem without intermediate constraint, comparison with other investigators yields excellent agreement. For the problem with intermediate constraint, results are presented for a wide variety of constraint positions and types, boundary conditions and circumferential mode numbers.