Continuous Limit of Multiple Gravitational Lens Effect and Average Magnification Factor

We show that the gravitational magnification factor averaged over all configurations of lenses in a locally inhomogeneous universe satisfy a second order differential equation with redshift $z$ by taking the continuous limit of multi-plane gravitational lens equation (the number $N$ of lenses $\to \mathrm{\infty }$) and that the gravitationally magnified Dyer-Roeder distance in a clumpy universe becomes to that of the Friedmann-Lema^{\i}tre universe for arbitrary values of the density parameter ${\mathrm{\Omega }}_0$ and of a mass fraction $\bar{\alpha }$ (smoothness parameter).