  A general equation for the centroid motion of free, continuous, intense beams
propagating off-axis in solenoidal periodic focusing fields is derived. The
centroid equation is found to be independent of the specific beam distribution
and may exhibit unstable solutions. A new Vlasov equilibrium for off-axis beam
propagation is also obtained. Properties of the equilibrium and the relevance
of centroid motion to beam confinement are discussed.
