  If $M$ is a projective manifold in $P^N$, then one can associate to each one
parameter subgroup $H$ of $SL(N+1)$ the Mumford $\mu$ invariant. The manifold
$M$ is Chow-Mumford stable if $\mu$ is positive for all $H$. Tian has defined
the notion of K-stability, and has shown it to be intimately related to the
existence of K\"ahler-Einstein metrics. The manifold $M$ is K-stable if $\mu'$
is positive for all $H$, where $\mu'$ is an invariant which is defined in terms
of the Mabuchi K-energy. In this paper we derive an explicit formula for $\mu'$
in the case where $M$ is a curve. The formula is similar to Mumford's formula
for $\mu$, and is likewise expressed in terms of the vertices of the Newton
diagram of a basis of holomorphic sections for the hyperplane line bundle.
