  A global analysis is presented of solutions for Laplace's equation on
three-dimensional Euclidean space in one of the most general orthogonal
asymmetric confocal cyclidic coordinate systems which admit solutions through
separation of variables. We refer to this coordinate system as five-cyclide
coordinates since the coordinate surfaces are given by two cyclides of genus
zero which represent the inversion at the unit sphere of each other, a cyclide
of genus one, and two disconnected cyclides of genus zero. This coordinate
system is obtained by stereographic projection of sphero-conal coordinates on
four-dimensional Euclidean space. The harmonics in this coordinate system are
given by products of solutions of second-order Fuchsian ordinary differential
equations with five elementary singularities. The Dirichlet problem for the
global harmonics in this coordinate system is solved using multiparameter
spectral theory in the regions bounded by the asymmetric confocal cyclidic
coordinate surfaces.
