  We present a new method for recovering the underlying velocity field from an
observed distribution of galaxies in redshift space. The method is based on a
kinematic Zel'dovich relation between the velocity and density fields in
redshift space. This relation is expressed in a differential equation slightly
modified from the usual Poisson equation, and which depends non-trivially on
$\beta \equiv \Omega^{0.6}/b$. The linear equation can be readily solved by
standard techniques of separation of variables by means of spherical harmonics.
One can also include a term describing the ``rocket effect" discussed by Kaiser
(1987). From this redshift space information alone, one can generate a
prediction of the peculiar velocity field for each harmonic ($l,m$) as a
function of distance. We note that for the quadrupole and higher order moments,
the equation is a boundary value problem with solutions dependent on both the
interior and exterior mass distribution. However, for a shell at distance $r$,
the dipole, as well as the monopole, of the velocity field in the Local Group
frame is fully determined by the interior mass distribution. This implies that
the shear of the measured velocity field, when fit to a dipole distortion,
should be aligned and consistent with the gravity field inferred from the well
determined local galaxy distribution. As a preliminary application we compute
the velocity dipole of distant shells as predicted from the 1.2Jy IRAS survey
compared to the measured velocity dipole on shells, as inferred from a recent
POTENT analysis. The coherence between the two fields is good, yielding a best
estimate of $\beta= 0.6 \pm 0.2$.
