  We consider the shape of the magnetic Compton profile (MCP), $J_{mag}(p_z)$,
in La$_{1.2}$Sr$_{1.8}$Mn$_2$O$_7$ for momentum transfer $p_z$ along the [110]
direction and the associated reciprocal form factor $B(r)$ defined by taking
the one-dimensional Fourier transform of $J_{mag}(p_z)$. $B(r)$ is shown to
contain a prominent dip at $r\approx$ 1 ${\AA}$, where the minimum value
$B_{min}$ of $B(r)$ can be related to the occupancies of the $e_g$ orbitals of
$d_{x^2-y^2}$ and $d_{3z^2-r^2}$ symmetry in the system. We illustrate our
procedure in detail by analyzing the measured MCP at 5K and the MCP computed
within the framework of the local spin density approximation (LSDA) and comment
on the differences between the measured and computed $e_g$ occupancies as a
reflection of the limitations of the LSDA in treating electron correlation
effects.
