  In this paper we study surfaces in Euclidean 3-space that satisfy a
Weingarten condition of linear type as $\kappa_1=m \kappa_2 +n$, where $m$ and
$n$ are real numbers and $\kappa_1$ and $\kappa_2$ denote the principal
curvatures at each point of the surface. We investigate the possible existence
of such surfaces parametrized by a uniparametric family of circles. Besides the
surfaces of revolution, we prove that not exist more except the case
$(m,n)=(-1,0)$, that is, if the surface is one of the classical examples of
minimal surfaces discovered by Riemann.
