${{\mbox{\boldmath$\sf C$}}^{1,p}}_{{{\rm B}}\to{{\rm A}}}$, respectively.

Gravity Between Well-Separated Cells {#sec:hard}
------------------------------------

In our implementation, we stick to a third order expansion ($p=3)$, whereby ignoring octopoles ${{\mbox{\boldmath$\sf M$}}^{3}}_{{\rm B}}$ (see remark above). The dipole ${{\mbox{\boldmath$\sf M$}}^{1}}_{{\rm B}}$ vanishes by construction and the Taylor-series coefficients for spherical Greens functions read (with Einstein’s sum convention) $$\label{hard-coeffs}
  \begin{array}{lcl}
    {{\sf C}^{0}}_{{{\rm B}}\to{{\rm A}}}        &=& {{\sf M}^{}}_{{\rm B}}\big[D^0
                                  + {{\textstyle{1\over2}}}{\tilde{\sf M}{}^{2}}_{{{\rm B}}ii}\,D^1
                                  + {{\textstyle{1\over2}}}R_iR_j{\tilde{\sf M}{}^{2}}_{{{\rm B}}ij}D^2
                                    \big],                           \\[1ex] 
    {{\sf C}^{1}}_{{{\rm B}}\to{{\rm A}},\,i}    &=& {{\sf M}^{}}_{{\rm B}}\big[R_i\big(D^1      
                                  + {{\textstyle{1\over2}}}{\tilde{\sf M}{}^{2}}_{{{\rm B}}jj}D^2
                                  + {{\textstyle{1\over2}}}R_jR_k{\tilde{\sf M}{}^{2}}_{{{\rm B}}jk}D^3\big)
                                  + R_j{\tilde{\sf M}{}^{2}}_{{{\rm B}}ij}D^2
                                    \big],                           \\[1ex]
    {{\sf C}^{2}}_{{{\rm B}}\to{{\rm A}},\,ij}   &=& {{\sf M}^{}}_{{\rm B}}\big[
                                    \delta_{ij}\,D^1+ R_iR_j\,D^2
                                    \big],                           \\[1ex]
    {{\sf C}^{3}}_{{{\rm B}}\to{{\rm A}},\,ijk}  &=& {{\sf M}^{}}_{{\rm B}}\big[\big(
                                    \delta_{ij}R_k+\delta_{jk}R_i
                                   +\delta_{ki}R_j\big)D^2
                                   +R_iR_jR_kD^3\big],
  \end{array}$$ where ${{\sf M}^{}}_{{\rm B}}\equiv{{\sf M}^{0}}_{{\rm B}}$ is the mass of cell B and ${\tilde{\sf M}{}^{2}}_{{\rm B}}\equiv{{\sf M}^{2}}_{{\rm B}}/{{\sf M}^{0}}_{{\rm B}}$ its [*specific*]{} quadrupole moment, while $$\label{Dn}
  D^m\equiv \left.
    \left({1\over r}{\partial\over\partial r}\right)^m
    g(r) \right|_{r=|{\mbox{\boldmath$\scriptstyleR$}}|}.$$ In practice, we calculate the coefficients ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{{\rm B}}\to{{\rm A}}}
\equiv{{\sf M}^{}}_{{\rm A}}{{\mbox{\boldmath$\sf C$}}^{m}}_{{{\rm B}}\to{{\rm A}}}$, because these obey the symmetry relations ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{2}}_{{{\rm B}}\to{{\rm A}}}= {\tilde{{\mbox{\boldmath$\sf C$}}}{}^{2}}_{{{\rm A}}\to{{\rm B}}}$ and ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{3}}_{{{\rm B}}\to{{\rm A}}}=-{\tilde{{\mbox{\boldmath$\sf C$}}}{}^{3}}_{{{\rm A}}\to{{\rm B}}}$ (for $p=3$), which arise from the mutuality of gravity. Since we 