the oscillations of Maclaurin spheroids.

The equations of motion (\[eq:AS:EULER\_S\]) and (\[eq:AS:EULER\_N\]), written in a frame rotating with angular velocity ${\mbox{\boldmath $\omega$}}$ relative to some inertial coordinate reference system, can be combined in a single equation

$$\begin{aligned}
\rho_\alpha\left({\partial\over\partial t}+u_{\alpha,j}
{\partial\over\partial x_j}\right)  u_{\alpha,i}
&=&-{\partial p_\alpha\over
\partial x_i}-\rho_\alpha{\partial\phi\over\partial x_i}
+{1\over 2}\rho_\alpha{\partial\vert{\mbox{\boldmath $\omega$}}
{\mbox{\boldmath $\times$}}{\bf x}\vert^2
\over\partial x_i}\nonumber\\
&+&2\rho_\alpha\epsilon_{ilm}u_{\alpha,l}\Omega_m +F_{\alpha\beta,i},
\label{eq:AS:euler}\end{aligned}$$

where the subscript $\alpha\in\{S,N\}$ identifies the fluid component, and Latin subscripts denote coordinate directions; $\rho_\alpha$, $p_\alpha$, and ${\bf u}_\alpha$ are the density, pressure, and velocity of fluid $\alpha$. The two fluids are coupled to one another by mutual gravitational attraction and the mutual friction force ${\bf F}_{\alpha\beta}$ \[Eq. (\[eq:AS2.4\])\]. The gravitational potential $\phi$ is derived from $$\begin{aligned}
\nabla^2\phi=\nabla^2(\phi_S+\phi_N)=4\pi G[\rho_S({\bf x})+\rho_N({\bf x})];\end{aligned}$$ the individual fluid potentials $\phi_\alpha$ obey $\nabla^2\phi_\alpha=4\pi G\rho_\alpha$. For a normal-superfluid mixture the mutual friction force, written in components, is $$\begin{aligned}
F_{SN,i}=-\rho_S\omega_S\beta_{ij}(u_{S,j}-u_{N,j}),\end{aligned}$$ where the mutual friction tensor is $$\begin{aligned}
\beta_{ij}=\beta\delta_{ij}+\beta^\prime\epsilon_{ijm}\nu_m
+(\beta^{\prime\prime}-\beta)\nu_i\nu_j,
\label{eq:AS:betaijdef}\end{aligned}$$ with $\beta$, $\beta^{\prime}$ and $\beta^{\prime\prime}$ being the mutual friction coefficients, and ${\mbox{\boldmath $\omega$}}_S
={\mbox{\boldmath $\nu$}}\omega_S\equiv{\mbox{\boldmath $\nabla\times$}}{\bf u}_S$.

The Euler equation (\[eq:AS:euler\]) can be extended to include external gravitational sources, for example the tidal potential of an external point source of gravity acting on an ellipsoid (Roche ellipsoids). We will 