  Firstly, a systematic procedure is derived for obtaining three-dimensional
bound-state equations from four-dimensional ones. Unlike ``quasi-potential
approaches'' this procedure does not involve the use of delta-function
constraints on the relative four-momentum. In the absence of negative-energy
states, the kernels of the three-dimensional equations derived by this
technique may be represented as sums of time-ordered perturbation theory
diagrams. Consequently, such equations have two major advantages over
quasi-potential equations: they may easily be written down in any Lorentz
frame, and they include the meson-retardation effects present in the original
four-dimensional equation. Secondly, a simple four-dimensional equation with
the correct one-body limit is obtained by a reorganization of the generalized
ladder Bethe-Salpeter kernel. Thirdly, our approach to deriving
three-dimensional equations is applied to this four-dimensional equation, thus
yielding a retarded interaction for use in the three-dimensional bound-state
equation of Wallace and Mandelzweig. The resulting three-dimensional equation
has the correct one-body limit and may be systematically improved upon. The
quality of the three-dimensional equation, and our general technique for
deriving such equations, is then tested by calculating bound-state properties
in a scalar field theory using six different bound-state equations. It is found
that equations obtained using the method espoused here approximate the wave
functions obtained from their parent four-dimensional equations significantly
better than the corresponding quasi-potential equations do.
