  We derive the four-dimensional field equations for the induced metric and
scalar field on the world-volume of a 3-brane in a five-dimensional bulk with
Einstein gravity plus a self-interacting scalar field. We calculate the
effective four-dimensional gravitational constant and cosmological constant for
arbitrary forms of the brane tension and self-interaction potential for the
scalar field in the bulk. In addition to the canonical energy-momentum tensor
for the scalar field and ordinary matter on the brane, the effective
four-dimensional Einstein equations include terms due to the scalar field and
gravitational waves in the bulk. We present solutions corresponding to static
Minkowski brane worlds and also dynamical Friedmann-Robertson-Walker brane
world cosmologies. We discuss the induced coupling of the scalar field to
ordinary matter on the brane.
