In appendix \[app:proj\], we show that the corresponding projected surface mass density profile is that of a Singular Isothermal Ellipsoid (SIE) which, following @kormannetal94, we write as $$\Sigma(\tilde x, \tilde y) = \frac{\sqrt{q}\tilde \sigma_v^2}{2G}\frac{1}{\tilde x^2+q^2\tilde y^2}$$ where both $q$ and $\tilde \sigma_v$ are known functions of $q_1,\ q_2$ and, in the case of $\tilde \sigma_v$, of $N(q_1,q_2)\sigma_v^2$ (see Appendix \[app:proj\] for details). In the above expression, $\tilde \sigma$ and $q$ are the effective velocity dispersion and axis ratio respectively of the projected SIE profile. As shown by @kormannetal94, the lensing cross section for an SIE scales trivially with the Einstein radius $b$[^4] $$b= 4\pi\frac{\tilde\sigma_v^2}{c^2}\frac{D_lD_{ls}}{D_s}
\label{eq:erad}$$ of the profile. Consequently, the distribution of halo orientations for a lens sample, $\rho({{\mathbf{\hat n}}})=\sigma_B({{\mathbf{\hat n}}})/{\left\langle \sigma_B \right\rangle}$, is independent of the velocity dispersion $\sigma_v$ of the halo.

There is one last important element of the model that needs to be specified, namely the luminosity function of the sources being lensed. Here, we take the luminosity function to be a power law with slope of $-2$, which, while not exactly correct, is reasonably close to the slope of the luminosity function of CLASS lenses [@chae03; @mckeanetal07]. Moreover, this choice is ideally suited for numerical work 