  The peculiar velocities of galaxies distort the pattern of galaxy clustering
in redshift space, making the redshift space power spectrum anisotropic. In the
linear regime, the strength of this distortion depends only on the ratio $\beta
\equiv f(\Omega)/b \approx \Omega^{0.6}/b$, where $\Omega$ is the cosmological
density parameter and $b$ is the bias parameter. We derive a linear theory
estimator for $\beta$ based on the harmonic moments of the redshift space power
spectrum. Using N-body simulations, we examine the impact of non-linear
gravitational clustering on the power spectrum anisotropy and on our
$\beta$-estimator. Non-linear effects can be important out to wavelengths
$\lambda \sim 50$Mpc/h or larger; in most cases, they lower the quadrupole
moment of the power spectrum and thereby depress the estimate of $\beta$ below
the true value. With a sufficiently large redshift survey, the scaling of
non-linear effects may allow separate determinations of $\Omega$ and $b$.
  We describe a practical technique for measuring the anisotropy of the power
spectrum from galaxy redshift surveys, and we test the technique on mock
catalogues drawn from the N-body simulations. Preliminary application of our
methods to the 1.2 Jy IRAS galaxy survey yields $\beta_{est} \sim 0.3-0.4 $ at
wavelengths $\lambda \sim 30-40$Mpc/h . Non-linear effects remain important at
these scales, so this estimate of $\beta$ is probably lower than the true
value.
