For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real (when the expression makes sense):
$$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Show that, for all $t \in \mathbb{R}$, the application
$$\mathrm{N}_t : \left|\begin{array}{rll} D(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \mathrm{N}(x,y,t) \end{array}\right.$$
is harmonic.
One may use question II.B.3.