  We present a boundary integral method, and an accompanying boundary element
discretization, for solving boundary-value problems for the Laplace-Beltrami
operator on the surface of the unit sphere $\S$ in $\mathbb{R}^3$. We consider
a closed curve ${\cal C}$ on ${\cal S}$ which divides ${\cal S}$ into two parts
${\cal S}_1$ and ${\cal S}_2$. In particular, ${\cal C} = \partial {\cal S}_1$
is the boundary curve of ${\cal S}_1$. We are interested in solving a boundary
value problem for the Laplace-Beltrami operator in $\S_2$, with boundary data
prescribed on $\C$.
