  In the recent paper [J.\ Phys.\ A 44 (2011) 065203], we have arrived at the
closed-form expression for the Green's function for the partial differential
operator describing propagation of a scalar wave in an $N$-dimensional
($N\geqslant2$) Maxwell fish-eye medium. The derivation has been based on
unique transformation properties of the fish-eye wave equation under the
hyperspherical inversion. In this communication, we arrive at the same
expression for the fish-eye Green's function following a different route. The
alternative derivation we present here exploits the fact that there is a close
mathematical relationship, through the stereographic projection, between the
wavized fish-eye problem in $\mathbb{R}^{N}$ and the problem of propagation of
scalar waves over the surface of the $N$-dimensional hypersphere.
