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