Five Families of Ruled Surfaces Through a Line of a Rectilinear Congruence, III

Five families of ruled surfaces through a line of a rectilinear congruence obtained by the repeated application of the operation of forming the Jacobian from Sannia's quadratic forms f ≡ G_aβ du^a du^β and φ ≡ε _aβ du^a du^β were studied in detail by Ogura [7, 8], Hayashi [2], Ram Behari [9] and Mishra [4]; and those from Kummer's quadratic forms f ≡ G_aβ du^a du^β and θ ≡ μ_aβ du^a du^β by Mishra [5]. It is the purpose of the present paper to study the ruled surfaces obtained by the repeated application of the# operation of forming the Jacobian from the quadratic forms θ and φ.