section \[sec:summary\] summarizes our work and presents our conclusions.

Lens Biases Induced by Triaxiality {#sec:biases}
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We begin by deriving the basic expressions on which we rely to estimate the effects of halo triaxiality on the observed properties of lensing galaxies. In particular, we show that since the lensing cross section for triaxial lenses is in general not spherically symmetric, this implies that a population of randomly oriented halos produces a non-random lens population. Finally, we show that the induced non-randomness of the lensing halo population can alter the mean observational properties of these halos relative to the general halo population.

The Lensing Cross Section {#sec:cs}
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Let ${\mathbf{p}}$ be a set of parameters that characterizes the projected gravitational potential of a halo. For instance, ${\mathbf{p}}$ can be the Einstein radius of the lens, its ellipticity, and so on. Given a background source density $n_s(z_s)$ and a halo density $n_h({\mathbf{p}},z_h)$, and in the absence of a flux limit, the mean number of lensing events per unit redshift per area is given by $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_s(z_s) n_h({\mathbf{p}},z_h)
\frac{d\chi}{dz_s}\frac{d\chi}{dz_h}\sigma({\mathbf{p}},z_h,z_s)$$ where $\chi$ is the comoving distance to the appropriate halo or source redshift, and $$\sigma({\mathbf{p}},z_h,z_s) = \int_{lensing} d^2{\mathbf{y}}.Ä
\label{eq:cs}$$ The integral is over 