In the first part we give a short proof for an old result by Beltrami and Dini on ruled surfaces satisfying a nontrivial relation between \(H\) and \(K\). In the second part we solve the analogous problem for the case of a relation between \(H\) and the inner curvature \(K_{II}\) of the second fundamental form. It turns out that in this case there are nontrivial examples which are not among the classical \(W\)-surfaces.