redshift by a halo along a given line of sight is given by Eq. \[eq:bcs\]. For convenience, we define the halo and source surface densities $d\Sigma_h/d{\mathbf{P}}$ and $d\Sigma_s/dz_s$ via $$\begin{aligned}
\frac{d\Sigma_h}{d{\mathbf{P}}dz_h}\ & =\ & \frac{dn_h}{d{\mathbf{P}}}\frac{d\chi}{dz_h} \\
\frac{d\Sigma_s}{dz_s} \ & =\ & n_s(>L_{min})\frac{d\chi}{dz_s}.\end{aligned}$$ In terms of these surface densities, and assuming a randomly-oriented distribution of halos, the mean number of lenses per unit area as a function of their orientation ${{\mathbf{\hat n}}}$ is given by $$\frac{dN_{lenses}}{d{\mathbf{P}}d{{\mathbf{\hat n}}}dz_sdz_hd\Omega} = \frac{1}{2\pi}\frac{d\Sigma_s}{dz_s}\frac{d\Sigma_h}{d{\mathbf{P}}dz_h}
    \sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}}),z_h,z_s,\alpha).
\label{eq:numlens}$$ The prefactor of $1/(2\pi)$ arises from the fact that $dn_h/d{\mathbf{P}}d{{\mathbf{\hat n}}}= (dn_h/d{\mathbf{P}})/(2\pi)$ due to our assumption of randomly oriented halos.[^2] We emphasize that Eq. \[eq:numlens\] characterizes the number of lenses *as a function of the relative orientation ${{\mathbf{\hat n}}}$ between the halo’s major axis and the line of sight. Thus, to compute the total number of lenses irrespective of halo orientation, we would simply integrate the above expression over all lines of sight ${{\mathbf{\hat n}}}$.*

There is an absolutely key point to be made concerning Eq. \[eq:numlens\], which provides the motivation behind this work. Specifically, we note that the number of lenses is proportional to $\sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}}))$. This implies that even though the overall halo population does not have 