the octants of the new root cell [@BH], and the total number of interactions rises from $N_I$ to $8N_I+N_+$, where $N_+$ interactions are needed for the mutual forces between these octants. In terms of a differential equation, this gives $$\label{ode}
        {{\rm d} N_I\over{\rm d} N}
        \simeq  {N_I\over N} {\Delta\ln N_I\over\Delta\ln N}
        \approx {N_I\over N} + {N_+\over N8\ln 8},$$ where the first term on the right-hand side accounts for the [ *intra-domain*]{} and the second for the [*inter-domain*]{} interactions. [Equation [(\[ode\])]{}]{} has solution $$\label{sol}
        N_I \simeq c_0 N + {N\over8\ln8} \int {N_+\over N^2}\;{\rm d} N.$$ In the tree code, every body requires a constant number of additional interactions, i.e. $N_+\propto N$, and the second term in [(\[sol\])]{} becomes $\propto N\ln N$ dominating $N_I(N)$. However, in the new code, $N_+$ grows sub-linear for large $N$, since a constant number of cell-cell interactions accounts for most new interactions of all bodies. In this case, the second term on the right-hand side of [equation [(\[sol\])]{}]{} also grows sub-linear with $N$ and $N_I$ will eventually be dominated by the first term. That is, in contrast to the tree code, the inter-domain interactions are neglible at large $N$ when compared to the intra-domain interactions. The transition 