  Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let
$\L=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special
geometry of $X$ implies that some sheaves related to the principal part bundles
of $\L$ are locally free. The inflectional loci of $X$ can be expressed in
terms of these sheaves, leading to explicit formulas for the cohomology classes
of the loci. The formulas imply that the only uninflected scrolls are the
balanced rational normal scrolls.
