B}}} \mu_i\,({{\mbox{\boldmath$y$}}}_i-{{\mbox{\boldmath$z$}}}_{{\rm B}})^{(n)}.\end{aligned}$$ The summation over $m$ in [equation [(\[local-expn\])]{}]{} represents the evaluation of gravity, represented by the field tensors ${{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}}$, at the evaluation point ${{\mbox{\boldmath$x$}}}$ within the sink cell A. The computation of the field tensors via the summation over $n$ in [equation [(\[coeffs\])]{}]{} represents the interaction between sink cell A and source cell B, represented by its multipole moments ${{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}}$.

The symmetry between  and  of the Taylor expansion [(\[Taylor\])]{} has the important consequence that, if [equations [(\[local-expn\])]{}]{} and [(\[coeffs\])]{} are used to compute ${\mbox{\boldmath$\nabla$}}\Phi_{{{\rm B}}\to{{\rm A}}}$ [*and*]{} ${\mbox{\boldmath$\nabla$}}\Phi_{{{\rm A}}\to{{\rm B}}}$, Newton’s third law is satisfied by construction. For instance, the sum over all forces of $N$ bodies vanishes within floating point accuracy.

Note that the highest-order multipole moments, ${{\mbox{\boldmath$\sf M$}}^{n=p}}$, contribute only to the coefficients ${{\mbox{\boldmath$\sf C$}}^{0,p}}$, and hence affect only $\Phi$ but not ${\mbox{\boldmath$\nabla$}}\Phi$. Since, in stellar dynamics the acceleration rather than the potential is to be computed, one may well ignore ${{\mbox{\boldmath$\sf M$}}^{p}}$, reducing CPU-time and memory requirements.

The formulae used in the standard tree code can be obtained by setting ${{\mbox{\boldmath$x$}}}={{\mbox{\boldmath$z$}}}_{{\rm A}}$, corresponding to body sinks. In this case, potential and acceleration are approximated by $-{{\mbox{\boldmath$\sf C$}}^{0,p}}_{{{\rm B}}\to{{\rm A}}}$ and 