  We show how to reduce the problem of symplectically embedding one
4-dimensional rational ellipsoid into another to a problem of embedding
disjoint unions of balls into appropriate blow ups of \C P^2. For example, the
problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that
of embedding k disjoint equal balls into \C P^2, and so can be solved by the
work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area
of the major axis to that of the minor axis.) As a consequence we show that the
ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k\ge 9, thus
answering a question raised by Hofer.
