  The subsonic, compressible, potential flow around a hypersphere can be
derived using the Janzen-Rayleigh expansion (JRE) of the flow potential in even
powers of the incident Mach number $\mathcal{M}_\infty$. JREs were carried out
with terms polynomial in the inverse radius $r^{-1}$ to high orders in two
dimensions (2D), but were limited to order $\mathcal{M}_\infty^4$ in three
dimensions (3D). We derive general JRE formulae to arbitrary order, adiabatic
index, and dimension. We find that powers of $\ln(r)$ can creep into the
expansion, and are essential in 3D beyond order $\mathcal{M}_\infty^4$. Such
terms are apparently absent in the 2D disk, as we confirm up to order
$\mathcal{M}_\infty^{100}$, although they do show in other dimensions (e.g. at
order $\mathcal{M}_\infty^2$ in 4D) and in non-circular 2D bodies. This
suggests that the disk, which was extensively used to study basic flow
properties, has additional symmetry. Our results are used to improve the
hodograph-based approximation for the flow in front of a sphere. The
symmetry-axis velocity profiles of axisymmetric flows around different prolate
spheroids are approximately related to each other by a simple, Mach-independent
scaling.
