  The spheroidal harmonics $S_{lm}(\theta;c)$ have attracted the attention of
both physicists and mathematicians over the years. These special functions play
a central role in the mathematical description of diverse physical phenomena,
including black-hole perturbation theory and wave scattering by nonspherical
objects. The asymptotic eigenvalues $\{A_{lm}(c)\}$ of these functions have
been determined by many authors. However, it should be emphasized that all
previous asymptotic analyzes were restricted either to the regime $m\to\infty$
with a fixed value of $c$, or to the complementary regime $|c|\to\infty$ with a
fixed value of $m$. A fuller understanding of the asymptotic behavior of the
eigenvalue spectrum requires an analysis which is asymptotically uniform in
both $m$ and $c$. In this paper we analyze the asymptotic eigenvalue spectrum
of these important functions in the double limit $m\to\infty$ and
$|c|\to\infty$ with a fixed $m/c$ ratio.
