  We derive semiclassical contributions of periodic orbits from a boundary
integral equation for three-dimensional billiard systems. We use an iterative
method that keeps track of the composition of the stability matrix and the
Maslov index as an orbit is traversed. Results are given for isolated periodic
orbits and rotationally invariant families of periodic orbits in axially
symmetric billiard systems. A practical method for determining the stability
matrix and the Maslov index is described.
