consider always both directions of any interaction, exploiting these relations substantially reduces the operational effort.

Accumulating Taylor Coefficients {#sec:accum}
--------------------------------

After, for each cell, the coefficients of all its interactions have been accumulated as ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{\rm A}}= \sum_{{\rm B}}{\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{{\rm B}}\to{{\rm A}}}$, where the sum includes all interaction partners B of cell A, we transform back to ${{\mbox{\boldmath$\sf C$}}^{m}}_{{\rm A}}={\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{\rm A}}/{{\sf M}^{}}_{{\rm A}}$. Next, for each body, the Taylor series of all relevant cells (those that contain the body), have to be accumulated by first translating to a common expansion center and then adding coefficients.

In contrast to expansions in spherical harmonics, the translation of the Cartesian expansion [(\[local-expn\])]{} to a different center ${{\mbox{\boldmath$z$}}}$ is straightforward. Let ${{\mbox{\boldmath$\sf C$}}^{m,p}}_0$ be the coefficients for expansion center ${{\mbox{\boldmath$z$}}}_0$, then the coefficients for expansion center ${{\mbox{\boldmath$z$}}}_1$ are $$\label{translate}
  {{\mbox{\boldmath$\sf C$}}^{m,p}}_1 = \sum_{n=0}^{p-m} {1\over n!}\,({{\mbox{\boldmath$z$}}}_0-{{\mbox{\boldmath$z$}}}_1)^{(n)} \odot
  {{\mbox{\boldmath$\sf C$}}^{m+n,p}}_0.$$

The Multipole Acceptance Criterion (MAC) {#sec:mac}
----------------------------------------

For the expansion [(\[local-expn\])]{} to converge, we must have $|{{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}}|
<|{{\mbox{\boldmath$R$}}}|$ for all body-body interactions ‘caught’ by a single cell-cell (or cell-body) interaction (see Appendix A). In order to ensure this, we first obtain, for each node, an upper limit $r_{\rm max}$ for the distance of any body 