A numerical method is presented for solving the equations of motion for axially symmetric flows of an incompressible, inviscid but rotational fluid. The method of solution is to replace derivatives in the axial direction by finite differences and then to solve the resulting set of ordinary non-linear differential equations by the Runge-Kutta method. Appropriate boundary conditions are given at some upstream cross-section and on some outer bounding surface. Along the axis of symmetry the condition of zero radial velocity is imposed. Difficulties associated with stability and with the two-point boundary conditions for the ordinary non-linear differential equations are described, and a sample calculation is presented.