The Five Families of Ruled Surfaces through a Line of a Rectilinear congruence

Ogura⁽¹⁾, Hayashi⁽²⁾) and Ram Behari⁽³⁾ have proved by different methods that repeated applications of the operation of forming the Jacobian from Sannia's fundamental quadratic forms⁽⁴⁾

f = Edu²+2 Fdu dv+G dv²

φ = δ du²+2 δ' du dv+δ" dv²

lead to only five distinct families of ruled surfaces through the line / of the congruence, viz.