$\psi$ as given we have, combining equations (\[expandedns\],45) and our definitions , $$\nabla^4 \psi - R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)} = \Gamma = \Gamma_{HA}+\Gamma_{NL}\ ,$$ where $$\Gamma_{HA} = \epsilon^4 \sin\theta g_1   =  \epsilon^4 \sin\theta \frac{2 K_1(\epsilon r)}{\epsilon K_0(\epsilon)} =\gamma_{HA}(\epsilon,r)  \sin\theta$$ and $$\begin{split}
\Gamma_{NL} &=  - R_e \epsilon^2 \sin\theta \cos\theta J(\epsilon, r)\\ &=- R_e \epsilon^2 \sin\theta \cos\theta  \frac{2 \left(K_0(\epsilon ) K_2(r \epsilon )-\frac{1}{r^2}K_0(r \epsilon ) K_2(\epsilon)\right)}{K_0(\epsilon )^2}\\
&= \gamma_{NL}(\epsilon, r)\sin\theta \cos\theta 
\end{split}$$ where these equations also serve to define the radial terms $\gamma_{XY}$. From now on we will work in terms of these radial functions, so it is useful just to write down their explicit forms: $$\gamma_{HA}(\epsilon,r) = 2 \epsilon^3 \frac{K_1(\epsilon r)}{K_0(\epsilon)}$$ $$\gamma_{NL}(\epsilon,r) = -2 R_e \epsilon^2 \frac{ \left(K_0(\epsilon ) K_2(r \epsilon )-\frac{1}{r^2}K_0(r \epsilon ) K_2(\epsilon)\right)}{K_0(\epsilon )^2}$$

Development of the iteration
----------------------------

We now attempt to refine the solution by writing the total solution as $$\Psi = \psi + \psi_2$$ Ideally we would like to arrange that $$0 = \nabla^4 \Psi - R_e \frac{\partial(\Psi, \delsq \Psi)}{\partial(x, y)}$$ The right side of this is given by the expansion $$\nabla^4 \psi - R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)}+\nabla^4 \psi_2 - R_e \frac{\partial(\psi_2, \delsq \psi)}{\partial(x, y)}- R_e \frac{\partial(\psi, \delsq \psi_2)}{\partial(x, y)}
- R_e \frac{\partial(\psi_2, \delsq 