variations of the new terms corresponding to the mutual gravitational attraction and mutual friction. The variation of the first force is $$\begin{aligned}
-\delta\int_{V_\alpha}{d^3x\rho_\alpha{\partial\phi_\beta
\over\partial x_i}}&=&
-\int_{V_\alpha}{d^3x\rho_\alpha({\bf x})
\xi_{\alpha,l}({\bf x}){\partial\over\partial x_l}}
\int_{V_\beta}{d^3x^{\prime}\rho_\beta({\bf x}^{\prime})(x_i-x^{\prime}_i)
\over\vert{\bf x}-{\bf x}^{\prime}\vert^3}
\nonumber\\
&+&\int_{V_\beta}{d^3x\rho_\beta({\bf x})\xi_{\beta,l}({\bf x}){\partial\over\partial x_l}}
\int_{V_\alpha}{d^3x^{\prime}\rho_\alpha({\bf x}^{\prime})(x_i-x^{\prime}_i)
\over\vert{\bf x}-{\bf x}^{\prime}\vert^3}
\label{eq:ASgravab}\end{aligned}$$ which is manifestly antisymmetric on $\alpha\leftrightarrow\beta$. Assuming $V_\alpha=V_\beta=V$ and $\rho_\alpha=f_\alpha\rho({\bf x})$ in the background equilibrium, we can simplify this to $$\begin{aligned}
-\delta\int_{V_\alpha}
d^3x\rho_\alpha\frac{\partial\phi_\beta}
{\partial x_i}&=&f_\alpha f_\beta
\int_V d^3x\rho({\bf x})[\xi_{\beta,l}
({\bf x})-\xi_{\alpha,l}({\bf x})]\nonumber\\
&\times&\frac{\partial}{\partial x_l}
\int_V\frac{d^2x^{\prime}\rho({\bf x}^{\prime})(x_i-x^{\prime}_i)}
{\vert{\bf x}-{\bf x}^{\prime}\vert^3}.\end{aligned}$$ Although we simplified the final answer by assuming that the fluids occupy identical volumes and have proportional densities in the background state, we could not have derived the correct perturbation of the first order virial theorem if we had not allowed the volumes to differ.

For uniform ellipsoids, we can simplify the mutual gravitational term further. First, we note that the second integral is simply the derivative of the gravitational potential which at any interior point of a homogeneous ellipsoid is $$\begin{aligned}
\label{eq:AS:ellips_phi}
\phi({\bf x})=-\pi G\rho\biggl(I-\sum_{k=1}^3A_kx_k^2\biggr),\end{aligned}$$ where $I$ is a constant. Thus, the mutual gravitational contribution to the equation of motion for the perturbed center-of-mass is $$\begin{aligned}
2\pi G\rho^2A_if_\alpha f_\beta 
\int_V{d^3x(\xi_{\beta,i}-\xi_{\alpha,i})}.\end{aligned}$$ If in the background 