This paper presents a transformation of the general time-dependent Euler equations for inviscid fluid flow to an inverse form, using two stream functions and the natural coordinate as independent variables. As a special case, the equations for axisymmetric flow are extended to compressible flow and also transformed to their inverse form. This way enables to introduce a theoretical basis for a class of inverse computational methods. Instead of searching for a geometry of a flow device that will lead to certain physical properties, the physical properties can be defined in advance, and the solution of the inverse equations directly leads to the corresponding geometry. So, it has become easier to arrive at an optimized design. However, the difficulty of defining an optimum remains.