1994A&A...287..349S

After summarizing the propagation equations for thin light beams in a general spacetime, as formulated in an accompanying paper (Seitz et al. 1994), we specialize them to the case of a clumpy universe, in which a fraction {tilde alpha} of the mean mass density is distributed homogeneously, whereas the rest is contained in spatially separated isolated clumps. The multiple deflection gravitational lens equation can then be obtained from a discretization of the general propagation equation. We investigate the behaviour of light beams between consecutive lens planes. In particular, a differential equation for the cross section of light beams is derived and solved, in terms of solutions of the Dyer-Roeder equation, which are derived for arbitrary {tilde alpha}≤1 and {OMEGA} in an Appendix. We then provide an alternative formulation of the magnification theorem in multiple deflection gravitational lens theory: a light beam which has not passed through a conjugate point cannot be demagnified relative to a reference beam, the so-called empty cone beam. This formulation is equivalent to the one given in Seitz & Schneider (1992), which states that a light beam which corresponds to a (local) minimum of the light travel time from the source to the observer cannot be demagnified relative to an empty cone beam; the proof given here is considerably simpler than the one in Seitz & Schneider. Finally, we show that the non-linear telescope matrix, introduced by Kovner (1987a) in the generalized quadrupole lens equation, is manifestly symmetric, so that the generalized quadrupole lens equation is formally equivalent to the ordinary quadrupole lens equation for a single deflector.