  The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing
functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs
equations for families of elliptic curves and associated surfaces. The
inhomogeneous Picard-Fuchs equations associated to elliptic integrals with
varying endpoints are derived and used to determine solutions of equations that
are algebraically related to a class of Painlev\'e VI equations.
