This paper describes the derivation of an optimum inlet Mach Number for the first stagesof an axial-flow compressor based on free vortex design. The optimum Mach Number is related to the given inlet conditions, dimensions, and tip speed and specifies the maximum amount of work tha t can be done on the fluid and the corresponding free vortex velocity diagrams at any radius in a stage where Mach Number effects are critical. A graphical method is described for simplifying the construction of free vortex velocity triangles for a stage at any radius of flow. This method is derived from the definition of free vortex flow. I t is shown that it is possible to design several successive stages using the same rotor blade form and the same stator blade form by merely changing the spacing and root stagger angle. This may be done without compromising the desired load distribution among stages and without violating the laws of free vortex flow.