Definition: A ruled surface is a furface swept out by a straight line $L$ moving along a curve $\beta$ . $$x(u,v)=\beta(u)+v \delta(u)$$ The definition above was in Barrett O'Neill's Element Differential Geometry . And there is a concept 'doubly ruled surface' in his exercise . However , the author did not define this concept before . If it means we can find two pair of function $(\beta_1,\delta_1)$ and $(\beta_2,\delta_2)$ such that $x(u,v)=\beta_i(u)+v \delta_i(u)$ $(i=1,2)$ denote the same surface . This seems not what we want to define , since for each surface of the form $$x(u,v)=\beta(u)+v \delta(u)$$ we can let $\beta_2=\beta$ and $\delta_2=\frac{\delta}{2}$ . So I think the definition of doubly ruled surface might be one of the following . But I'm not sure what it really means.
(1) $(\beta_1,\delta_1)$ and $(\beta_2,\delta_2)$ such that the image of $\beta_1(U)$ and $\beta_2(U)$ are not the same .
(2) $(\beta_1,\delta_1)$ and $(\beta_2,\delta_2)$ such that $\beta_1=\beta_2$ and $\delta_1(u) \neq k(u) \delta_2(u)$
We say that a surface is doubly ruled when, through each of its points, there are two distinct lines that lie on the surface.