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where r, 6 are polar coordinates, M is the Mach number of the flow at some distance from the origin where the velocity is assumed to be uniform, y is the ratio of specific heats of the fluid and <p, q are the velocity potential and velocity, respectively, at the point r, 0. All distances are referred to some typical dimension in the field, and velocities are referred to the speed of the uniform stream. The boundary conditions to be imposed on <p are (1) d<p/dr-+cos 6 as r—» °°, where 6 is measured from the direction of the uniform stream, and (2) the normal derivative of <p equals zero at the boundary of a fixed obstacle in the stream. Many attempts have been made to solve this equation analytically, but none has met with complete success. Most of the progress has been through some linearizing process, such as writing <p as a power series in M2, the non-linear terms on the right hand side of (1.1) then being of higher order in M2 than the linear terms on the left. Again, a transformation to new independent variables, the magnitude and direction of the local fluid velocity, gives a linear equation for <f>. In this paper, use is made of an expansion of <p as a power series in r~l and it is shown that the equation for each of the coefficients in the series is linear. Some light is thrown on the form of the expression for <p which may be expected in the two cases of zero and finite circulation about the obstacle. 2. Case of zero circulation. For the case of flow past an obstacle about which there is no circulation we make the assumption that <p can be written in the following form:

Power series expansions of the velocity potential in compressible flow