where $\mu_i$ is the weight of the $i$th body.

Consider two cells A and B, see Figure \[fig:cells\], with centers of mass ${{\mbox{\boldmath$z$}}}_{{\rm A}}$ and ${{\mbox{\boldmath$z$}}}_{{\rm B}}$, respectively. The Greens function describing the mutual interaction between a body at position ${{\mbox{\boldmath$x$}}}$ in cell A and a body at position ${{\mbox{\boldmath$y$}}}$ in cell B may be Taylor expanded about the separation ${{\mbox{\boldmath$R$}}}\equiv{{\mbox{\boldmath$z$}}}_{{\rm A}}-{{\mbox{\boldmath$z$}}}_{{\rm B}}$: $$\label{Taylor}
  g({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}) = \sum_{n=0}^p {1\over n!}\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}})^{(n)}
  \odot{\mbox{\boldmath$\nabla$}}^{(n)}g({{\mbox{\boldmath$R$}}})\;+\;{\cal R}_p(g),$$ where $p$ is the order of the expansion, while ${\cal R}_p$ denotes the Taylor series remainder (see also Appendix A). Here, we follow Warren & Salmon [@WS] by using the notational shorthand in which ${{\mbox{\boldmath$x$}}}^{(n)}$ indicates the $n$-fold outer product of the vector ${{\mbox{\boldmath$x$}}}$ with itself, while $\odot$ denotes a tensor inner product. When inserting [(\[Taylor\])]{} into [(\[potential\])]{}, whereby restricting the sum over $j$ to all bodies within cell B, we obtain for the potential at every position ${{\mbox{\boldmath$x$}}}$ in cell A and generated by all bodies in cell B [@WS] $$\begin{aligned}
 \label{local-expn}
  \Phi_{{{\rm B}}\to{{\rm A}}}({{\mbox{\boldmath$x$}}}) &=&-\sum_{m=0}^p {1\over m!}\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$z$}}}_{{\rm A}})^{(m)} 
                            \odot{{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}}
                            \;+\;{\cal R}_p(\Phi_{{{\rm B}}\to{{\rm A}}})
                            \\[1ex] \label{coeffs}
  {{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}}  &=& \sum_{n=0}^{p-m} {(-1)^n\over n!}\;{\mbox{\boldmath$\nabla$}}^{(n+m)}
                            g({{\mbox{\boldmath$R$}}}) \odot {{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}},
                            \\[1ex] \label{multipoles}
  {{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}}&=& \sum_{{\mbox{\boldmath$\scriptstyley$}}_i\in{{\rm 