  We solve Einstein's constraint equations in the conformal thin-sandwich
decomposition to model thin shells of non-interacting particles in circular
orbit about a non-rotating black hole. We use these simple models to explore
the effects of some of the freely specifiable quantities in this decomposition
on the physical content of the solutions. Specifically, we adopt either maximal
slicing or Kerr-Schild slicing, and make different choices for the value of the
lapse on the black hole horizon. For one particular choice of these quantities
the resulting equations can be solved analytically; for all others we construct
numerical solutions. We find that these different choices have no effect on our
solutions when they are expressed in terms of gauge-invariant quantities.
