In this paper I present a new representation of the multipole expansion used in tree algorithm and FMM. In the new method, gravitational field due to the multipole expansion is approximated by the potential due to a set of pseudoparticles on a ring or a sphere.
The new method is quite similar to Anderson's method, which uses the value of potential itself on a ring. However, compared to Anderson's original algorithm, the new method is more accurate for the same number of points.
On a general-purpose computer, the performance of the new method and Anderson's method would be practically the same. However, when combined with special-purpose computers, the new method offers a huge advantage, since the evaluation of the multipole expansion can be done on a hardware which is specialized to the calculation of the interaction of point particles.