  In this note we show that the periodic b-equation can only be realized as an
Euler equation on the Lie group Diff(S^1) of all smooth and orientiation
preserving diffeomorphisms on the cirlce if b=2, i.e. for the Camassa-Holm
equation. In this case the inertia operator generating the metric on Diff(S^1)
is given by A=1-d^2/dx^2. In contrast, the Degasperis-Procesi equation, for
which b=3, is not an Euler equation on Diff(S^1) for any inertia operator. Our
result generalizes a recent result of B. Kolev.
