\delta_2
\end{array}
\right)
\label{eq:Fsolve}$$ where ${\cal M}$ is a $4\times 4$ matrix consisting of elements proportional to sums of $Q_{ijkl}$ and $Q_{ij}Q_{kl}$, the former of which can be found by explicitly expanding the expressions in Okura et al., and the latter of which is again derived from the shift in the centroid. For the convenience of the reader, we write out the explicit form of ${\cal M}$ in Appendix A.

It may be seen by examining the elements of ${\cal M}$ why this inversion must be done explicitly for even mildly elliptical sources. For fully circular sources, it may be seen by inspection that ${\cal
M}$ is diagonal. However, when a source has an ellipticity even as small as $10\%$, it can be shown that $|M_{11}|\simeq |M_{12}|$, and thus equations (\[eq:skewness\]) and (\[eq:arciness\]) are no longer even approximately correct.

Gaussian Weighting with HOLICs
------------------------------

The application of the HOLICs technique would be trivial if there were no measurement noise. In the presence of noise, and especially, when the sky dominates, measurement of unweighted moments is inherently quite noisy. In a case where we are measuring the 3rd and 4th moments, it is even more so.

Kaiser, Squires & Broadhurst (1995; 