We consider Dirichlet eigenfunctions of membrane problems. A counterexample to Payne's nodal line conjecture is given, i.e. a domain in $\mathbb R^2$ (not simply connected) whose second eigenfunction has a nodal set disjoint from the boundary. Also a domain in $\mathbb R^2$ is given whose second eigenvalue has multiplicity three. Furthermore, some sufficient conditions are given which imply that an eigenfunction of a Dirichlet membrane problem in $\mathbb R^n$ has a zero set which hits the boundary.
( in: "Advances in differential equations and mathematical physics (Atlanta, GA, 1997) ", Amer. Math. Soc. , (1998), pp. 33-48,) 16 pages