  The multiple scattering formalism is used to extract irreducible N-body parts
of Green's functions and Casimir energies describing the interaction of N
objects that are not necessarily mutually disjoint. The irreducible N-body
scattering matrix is expressed in terms of single-body transition matrices. The
irreducible N-body Casimir energy is the trace of the corresponding irreducible
N-body part of the Green's function. This formalism requires the solution of a
set of linear integral equations. The irreducible three-body Green's function
and the corresponding Casimir energy of a massless scalar field interacting
with potentials are obtained and evaluated for three parallel semitransparent
plates. When Dirichlet boundary conditions are imposed on a plate the Green's
function and Casimir energy decouple into contributions from two disjoint
regions. We also consider weakly interacting triangular--and parabolic-wedges
placed atop a Dirichlet plate. The irreducible three-body Casimir energy of a
triangular--and parabolic-wedge is minimal when the shorter side of the wedge
is perpendicular to the Dirichlet plate. The irreducible three-body
contribution to the vacuum energy is finite and positive in all the cases
studied.
