We consider solutions $u_n (x,y)$ of Laplace’s equation which are regular in the interior of a smooth closed plane curve c, and the boundary conditions $\partial u_n /\partial v = \lambda _n gu_n $, where g is sufficiently smooth, positive, periodic and a prescribed function of arclength, and $u_n $, $\lambda _n $ are eigenfunctions and eigenvalues to be determined. For large $\lambda _n $ we show that $\lambda _n = O(n)$, n a large integer, and that $u_n $ is trigonometric, asymptotically. The method employed is the reduction of the problem to a boundary integral equation and the studying of that equation. The results are confirmed in a separable case which arises by a special choice of $g(s)$ and the curve c.