${{\mathcal S}}$, associated to $[\prod_{j=1}^p (x-\alpha_j)(x-\beta_j)]^{1/2}$. This surface has a natural projection $\pi :{{\mathcal S}}\to{{\mathbb{C}}}$, a twofold cover except at the branch points $\{\alpha_j\}_{j=1}^p \cup \{\beta_j\}_{j=1}^p$. $\pi^{-1} [\beta_j, \alpha_{j+1}]$ is a circle and $m(z)$ has exactly one pole $\gamma_1, \dots, \gamma_{p-1}$ on each circle.

It has been known for many years (see Faber [@Fab]) that the density of zeros $dk$ is supported on $\cup_{j=1}^p [\alpha_j, \beta_j]\equiv B$ and is the equilibrium measure for $B$ in potential theory. We define $k(E) = 
\int_{\alpha_1}^E dk$. Then $k(\beta_j)= j/p$. Our main results about OPRL are:

We can describe the zeros of $p_{np-1}(x)$ exactly (not just asymptotically) in terms of $\pi (\gamma_j)$ and $k(E)$.

Asymptotically, as $n\to\infty$, the number of zeros of $p_n$ in each band $[\alpha_j, \beta_j]$, $N^{(n,j)}$, obeys $\sup_n {\lvert{\frac}{n}{p} - N^{(n,j)}\rvert}<\infty$, and the zeros $\{x_\ell^{(n,j)}\}_{\ell=1}^{N(n,j)}$ obey $${\label}{1.8} 
\sup_{\substack{ j \\ \ell=1,2, \dots, N^{(n,j)}-1}} \, n\biggl| k(x_{\ell+1}^{(n,j)}) - 
k(x_\ell^{(n,j)}) - {\frac}{1}{n}\biggr| \to 0$$ as $n\to\infty$.

$z\in{{\mathbb{C}}}$ is a limit of zeros of $p_n$ if and only if $z$ lies in ${\text{\rm{supp}}}(d\rho)$.

Outside the bands, there are at most $2p+2b-3$ points which are limits of zeros of $p_{mp+b-1}$ for each $b=1, \dots, p$ and, except for these limits, zeros have no accumulation points 