  Let X be a smooth double cover of a geometrically ruled surface defined over
a separably closed field of characteristic different from 2. The main result of
this paper is a finite presentation of the 2-torsion in the Brauer group of X
with generators given by central simple algebras over the function field of X
and relations coming from the N\'eron-Severi group of X. In particular, the
result gives a central simple algebra representative for the unique nontrivial
Brauer class on any Enriques surface. An example demonstrating the applications
to the study of rational points is given.
