  Polynomial solutions to the generalized Lam\'e equation, the Stieltjes
polynomials, and the associated Van Vleck polynomials have been studied since
the 1830's in various contexts including the solution of Laplace equations on
an ellipsoid. Recently there has been renewed interest in the distribution of
the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes
polynomials increases. In this paper we show that the zeros of Van Vleck
polynomials corresponding to Stieltjes polynomials of successive degrees
interlace. We also show that the spectral polynomials formed from the Van Vleck
zeros are not orthogonal with respect to any weight. This furnishes a
counterexample, coming from a second order differential equation, to the
converse of the well known theorem that the zeros of orthogonal polynomials
interlace.
