In this paper methods are developed for the calculation of incompressible and compressible flow under conditions of axial symmetry. Relaxation is used to solve new forms of the flow equations. In these equations Φ, the velocity potential, and ψ, Stokes's stream function, for the incompressible flow, are the independent variables. The radial distance, r, is the dependent variable in the incompressible flow equation, and this feature renders the method especially suitable for the design of axially symmetric ducts. A typical design problem is calculated as an example at the end of the paper. Log(I/q), where q is the velocity, is the dependent variable in the compressible flow equation, which can only be solved after the incompressible flow solution has been found.