Neumann problems for Laplace's equation determine $u$ only up to a constant

Let $u_1$ and $u_2$ be solutions of $\Delta u=0$ in D, $\partial u/\partial n=g(s)$ on $\partial D$, where D is the ellipse $x^2/4+y^2/9\le1$. If $u_1(0,\tfrac12)-u_2(0,\tfrac12)=1$, then: