  We study quotients of quadratic forms and associated polar lines in the
projective plane. Our results, applied pointwise to quadratic differential
forms, shed some light on classical binary differential equations (BDEs)
associated to congruences of lines in Euclidean 3-space and allows us to
introduce a new one. The new BDE yields a new singular surface in the Euclidean
3-space associated to a congruence of lines. We determine the generic local
configurations of the above BDEs on congruences.
