  The aim of this paper is to introduce a new inversion procedure for re-
covering functions, defined on $\Bbb R^{2}$, from the spherical mean transform,
which integrates functions on a prescribed family $\Lambda$ of circles, where
$\Lambda$ consists of circles whose centers belong to a given ellipse E on the
plane. The method presented here follows the same procedure which was used by
S. J. Norton in [22] for recovering functions in case where $\Lambda$ consists
of circles with centers on a circle. However, at some point we will have to
modify the method in [22] by using expansion in elliptical coordinates, rather
than spherical coordinates, in order to solve the more generalized elliptical
case. We will rely on a recent result obtained by H.S. Cohl and H.Volkmer in
[8] for the eigenfunction expansion of the Bessel function in elliptical
coordinates.
