For the motions caused by the uniform expansion of a cylinder or sphere into still air, and for steady symmetrical supersonic flow past a cone, first approximations are obtained to the strength and position of the shock-wave for small disturbances. It is also shown what are the correct approximations to the pressure on the surface. The work is mathematically rigorous, and provides some justification of approximate methods that have been used, for example, in the general theory of supersonic flow past bodies of revolution.