centroid proportional to the quadrupole moments. In order to correctly invert the HOLICs, this term needs to be incorporated explicitly. The simplicity of the extra term results from an approximation of near circularity.

The beauty of this approach is that it gives us a very intuitive feel for what flexion means in an observational way. We thus introduce the term “skewness” to the intrinsic properties of a galaxy as measured from equation (\[eq:skewness\]) whether or not the galaxy is otherwise circular, and whether or not it is lensed. The skewness may be thought of as the intrinsic property, much as the “ellipticity” is the intrinsic property related to the “shear.” Likewise, the intrinsic property associated with equation (\[eq:arciness\]) will be referred to as the “arciness.”

In reality, however, equations (\[eq:skewness\]) and (\[eq:arciness\]) are not sufficient to perform a flexion estimate even if a galaxy has an ellipticity of only a few percent. Okura et al provide a general relationship between estimators for flexion and HOLICs, though the relation is best expressed in matrix form: $${\cal M} \left(
\begin{array}{c}
{\cal F}_1 \\
{\cal F}_2 \\ 
{\cal G}_1 \\
{\cal G}_2
\end{array}
\right)
=
\left(
\begin{array}{c}
\zeta_1 \\
\zeta_2 \\ 
\delta_1 \\
