  Two approaches for determining Hilbert functions of fat point subschemes of
$\mathbb P^2$ are demonstrated. A complete determination of the Hilbert
functions which occur for 9 double points is given using the first approach,
extending results obtained in a previous paper using the second approach. In
addition the second approach is used to obtain a complete determination of the
Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points
are smooth points of an irreducible plane cubic curve. Additional results are
obtained using the first approach for $n\geq 9$ double points when the points
lie on an irreducible cubic (but now are not assumed to be smooth points of the
cubic).
