A_\varphi \sim  {\cal O} \left(\frac{1}{r}\right) \ ,
   \hspace{10pt}(r \rightarrow \infty) \ .
   \label{Eq:BC_A}\end{aligned}$$ This boundary condition for $A_\varphi$ results in $$\begin{aligned}
   H_p \sim {\cal O} \left(\frac{1}{r^2}\right) \ ,
   \hspace{10pt}(r \rightarrow \infty) \ .
   \label{Eq:BC_H}\end{aligned}$$ where $H_p$ is the [*poloidal*]{} magnetic field. From these boundary conditions and using a proper Green’s function for the Laplacian, we have the integral representations of equation (\[Eq:Poisson\]) and equation (\[Eq:Poisson\_A\]) as follows: $$\phi_g(\Vec{r}) = -G \int \frac{\rho(\Vec{r}')}{|\Vec{r} - \Vec{r}'|}
   \, d^3 \Vec{r}' \ ,$$ $$A_\varphi(\Vec{r}) \sin \varphi
   = - \int \frac{S_A(\Vec{r}')\sin\varphi'}
   {|\Vec{r} - \Vec{r}'|} \, d^3 \Vec{r}' \ .$$ Therefore, we can obtain smooth potentials, $\phi_g$ and $A_\varphi$ by integrating these equations. Since we have chosen the functional form of the current density which decreases near the surface and vanishes at the stellar surface sufficiently smoothly, we obtain continuous [*poloidal*]{} magnetic fields from $A_\varphi$.

Global characteristics of equilibria
------------------------------------

To see the global characteristic of magnetized equilibria, we define some integrated quantities as follows: $$\begin{aligned}
   W \equiv \frac{1}{2}\int \phi_g \rho \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
   T \equiv \frac{1}{2} \int \rho (R \Omega)^2 \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
   \Pi \equiv \int p \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
   U \equiv N \Pi \ ,\end{aligned}$$ 