  For the random eigenvalues with density corresponding to the Jacobi ensemble
$$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 -
\lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i) $$ $(a, b > -1, \beta >
0) $ a strong uniform approximation by the roots of the Jacobi polynomials is
derived if the parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$.
Roughly speaking, the eigenvalues can be uniformly approximated by roots of
Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/\beta-1)$, where
the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to
investigate the asymptotic properties of the corresponding spectral
distribution if $n \to \infty$ and the parameters $a, b$ and $\beta$ vary with
$n$. We also discuss further applications in the context of multivariate random
$F$-matrices.
