state, the two fluids move together or are stationary, the variation of the mutual friction force becomes $$\begin{aligned}
\delta\int_V{d^3x F_{\alpha\beta,i}}=-{\cal S}_{\alpha\beta}
{d\over dt}\biggl[f_S\int_V{d^3x\rho({\bf x})\omega_S\beta_{ij}
(\xi_{S,j}-\xi_{N,j})}\biggr],\end{aligned}$$ where ${\cal S}_{\alpha\beta}=0$ if $\alpha=\beta$, 1 if $\alpha=S$ and $\beta=N$, and $-1$ if $\alpha=N$ and $\beta=S$. Collecting terms we find the first order virial equation $$\begin{aligned}
f_{\alpha} \frac{d^2 V_{\alpha,i} }{dt^2}&=&
2\epsilon_{ilm}f_{\alpha}\Omega_m\frac{d}{dt} V_{\alpha,l}
+(\Omega^2\delta_{ij} -\Omega_i\Omega_j) f_{\alpha} V_{\alpha,j}\nonumber\\
&-&2\pi G \rho A_i f_{\alpha} f_{\beta} \left(V_{\alpha,i}-
V_{\beta,i}\right)
-{\cal S}_{\alpha\beta}f_S\omega_S \beta_{ij}
\left(V_{\alpha,j}-V_{\beta,j}\right).
\label{eq:AS:PERTURB}\end{aligned}$$ The CM and relative motions can be decoupled by defining $$\begin{aligned}
V_{i}\equiv f_SV_{S,i}+f_NV_{N,i},\qquad
U_{i}\equiv V_{S,i}-V_{N,i}.
\label{eq:ASNEW_VIRIALS}\end{aligned}$$ The CM motions of two fluids are trivial (as they can be eliminated by a linear transformation of the reference frame) and, therefore, $V_i =0$. The virial equation describing the relative motions of the two fluids is $$\begin{aligned}
\frac{d^2}{dt^2}U_i = 2 \epsilon_{ilm}\Omega_m\frac{d}{dt} U_l
+(\Omega^2\delta_{ij}-\Omega_i\Omega_j) U_j - 2 A_i U_i
-2 \Omega \tilde \beta_{ij} \frac{d}{dt}U_j.
\label{eq:AS:monopole}\end{aligned}$$ From the latter equation it is straightforward to compute the first harmonic relative oscillation modes of irrotational ellipsoids, which is done in the next section for Maclaurin spheroids.

Second order virial equations
-----------------------------

Taking the first moment of Eq. (\[eq:AS:euler\]) results in the second order ‘virial’ equation $$\begin{aligned}
{d\over dt}\biggl(\int_{V_\alpha}{d^3x\rho_\alpha x_ju_{\alpha,i}}
\biggr)&=&2\epsilon_{ilm}\Omega_m\biggl(\int_{V_\alpha}
{d^3x\rho_\alpha x_ju_{\alpha,l}}\biggr)
+\Omega^2I_{\alpha,ij}-\Omega_i\Omega_kI_{\alpha,kj}
