  Perhaps the most powerful method for deriving the Newtonian gravitational
interaction between two masses is the multipole expansion. Once inner
multipoles are calculated for a particular shape this shape can be rotated,
translated, and even converted to an outer multipole with well established
methods. The most difficult stage of the multipole expansion is generating the
initial inner multipole moments without resorting to three dimensional
numerical integration of complex functions. Previous work has produced
expressions for the low degree inner multipoles for certain elementary solids.
This work goes further by presenting closed form expressions for all degrees
and orders. A combination of these solids, combined with the aforementioned
multipole transformations, can be used to model the complex structures often
used in precision gravitation experiments.
