  Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ be
$p \times p$ positive definite covariance matrices. On being given random
samples of sizes $N_1$ and $N_2$ from independent multivariate normal
populations $N_p(\mu,\Sigma_1)$ and $N_p(\mu,\Sigma_2)$, respectively, the
Behrens-Fisher problem is to solve the likelihood equations for estimating the
unknown parameters $\mu$, $\Sigma_1$, and $\Sigma_2$. We shall prove that for
$N_1, N_2 > p$ there are, almost surely, exactly $2p+1$ complex solutions of
the likelihood equations. For the case in which $p = 2$, we utilize Monte Carlo
simulation to estimate the relative frequency with which a typical
Behrens-Fisher problem has multiple real solutions; we find that multiple real
solutions occur infrequently.
