  For an algebraic function field $F/K$ and a discrete valuation $v$ of $K$
with perfect residue field $k$, we bound the number of discrete valuations on
$F$ extending $v$ whose residue fields are algebraic function fields of genus
zero over $k$ but not ruled. Assuming that $K$ is relatively algebraically
closed in $F$, we find that the number of nonruled residually transcendental
extensions of $v$ to $F$ is bounded by $\mathfrak{g}+1$ where $\mathfrak{g}$ is
the genus of $F/K$. An application to sums of squares in function fields of
curves over $\mathbb{R}(\!(t)\!)$ is presented.
