  By extending the concept of Euler-angle rotations to more than three
dimensions, we develop the systematics under rotations in higher-dimensional
space for a novel set of hyperspherical harmonics. Applying this formalism, we
determine all pairwise Coulomb interactions in a few-body system without
recourse to multipole expansions. Our approach combines the advantages of
relative coordinates with those of the hyperspherical description. In the
present method, each Coulomb matrix element reduces to the ``1/r'' form
familiar from the two-body problem. Consequently, our calculation accounts for
all the cusps in the wave function whenever an interparticle separation
vanishes. Unlike a truncated multipole expansion, the calculation presented
here is exact. Following the systematic development of the procedure for an
arbitrary number of particles, we demonstrate it explicitly with the simplest
nontrivial example, the three-body system.
