The seminal work of Nijboer and De Wette [Physica 23, 309 (1957)] enables the calculation of lattice sums of spherical harmonics, but has long been overlooked. In this article, their central result is recast in a simplified form suitable for modern multipole algorithms that employ the solid harmonics. This formulation makes possible the imposition of periodic boundary conditions within modern versions of the fast multipole method, and other fast N-body methods. The distinction between the extrinsic values obtained with the lattice sums M of the multipole interaction tensors, and the intrinsic values associated with Taylor's expansion of the Ewald formulas, is made. The central constants, M, are computed to 32 digit accuracy using extended precision arithmetic. Timings and corresponding errors obtained with a periodic version of the fast multipole method are presented for particle numbers spanning [10(3),10(6)], and a range of expansion orders. A qualitative comparison is made between the present implementation, other periodic versions of the fast multipole method, and fast Ewald methods. (C) 1997 American Institute of Physics. [S0021-9606(97)02947-4].