  The Fisher-Bingham distribution ($\mathrm{FB}_8$) is an eight-parameter
family of probability density functions (PDF) on $S^2$ that, under certain
conditions, reduce to spherical analogues of bivariate normal PDFs. Due to
difficulties in computing its overall normalization constant, applications have
been mainly restricted to subclasses of $\mathrm{FB}_8$, such as the Kent
($\mathrm{FB}_5$) or von Mises-Fisher (vMF) distributions. However, these
subclasses often do not adequately describe directional data that are not
symmetric along great circles. The normalizing constant of $\mathrm{FB}_8$ can
be numerically integrated, and recently Kume and Sei showed that it can be
computed using an adjusted holonomic gradient method. Both approaches, however,
can be computationally expensive. In this paper, I show that the normalization
of $\mathrm{FB}_8$ can be expressed as an infinite sum consisting of
hypergeometric functions, similar to that of the $\mathrm{FB}_5$. This allows
the normalization to be computed under summation with adequate stopping
conditions. I then fit the $\mathrm{FB}_8$ to a synthetic dataset using a
maximum-likelihood approach and show its improvements over a fit with the more
restrictive $\mathrm{FB}_5$ distribution.
