  Polynomial solutions to the generalized Lam\'e equation, the
\textit{Stieltjes polynomials}, and the associated \textit{Van Vleck
polynomials} have been studied since the 1830's, beginning with Lam\'e in his
studies of the Laplace equation on an ellipsoid, and in an ever widening
variety of applications since. In this paper we show how the zeros of Stieltjes
polynomials are distributed and present two new interlacing theorems. We
arrange the Stieltjes polynomials according to their Van Vleck zeros and show,
firstly, that the zeros of successive Stieltjes polynomials of the same degree
interlace, and secondly, that the zeros of Stieltjes polynomials of successive
degrees interlace. We use these results to deduce new asymptotic properties of
Stieltjes and Van Vleck polynomials. We also show that no sequence of Stieltjes
polynomials is orthogonal.
