Summary: The only case in which the motion of a gas at high speed in three dimensions has so far been discussed mathematically is that of the disturbance produced by a cone moving with velocity greater than that of sound. In the present work another case is analysed, namely, the radial outward flow produced by a uniformly expanding sphere. The region of expanding air is bounded by a shock wave outside which the air is undisturbed. As the radial velocity of the sphere increases the thickness of the layer of disturbed air decreases till at infinite rate of expansion it is only 6% of the radius of the sphere. The distributions of velocity and pressure are given for a range of rates of expansion. When the radial velocity of the sphere is small an approximate analysis based on the theory of sound yields results which are inaccurate near the sphere and also at the shock wave which forms the outer boundary of the expanding air.