## Abstract

The subsonic potential flow equations for a perfect gas are transformed by means of dependent variables s = ($\rho/\rho_0$)$^n$ q/a$_0$ and $\sigma$ = /frac{1}{2} ln ($\rho_0$/$\rho$), where q is the local velocity, $\rho$ and $\alpha$ the local density and speed of sound, and the suffix 0 indicates stagnation conditions. n is a parameter which is to be chosen to optimize the approximations. Bernoulli's equation then becomes a relation between s$^2$ and $\sigma$ which is independent of initial conditions. A family of first-approximation solutions in terms of the incompressible solution is obtained on linearizing. It is shown that for two-dimensional flow, the choice n = 0.5 gives results as accurate as those obtained with the Karman-Tsien solution. The exact equations are then transformed into the plane of the incompressible velocity potential and stream function and the first-approximation results substituted in the nonlinear terms. The resulting second-approximation equations can then be solved by a relaxation method and the error in this approximation estimated by carrying out the third-approximation solution. Results are given for a circular cylinder at a free-stream Mach number, M$_\infty$ = 0.4, and a sphere at M$_\infty$ = 0.5. The error in the velocity distribution is shown to be less than $\pm$ 1% in the two-dimensional case. A rough and ready compressibility rule is formulated for axisymmetric bodies, dependent on their thickness ratios.