  We measure the anisotropy of the redshift-space power spectrum in the 1.2-Jy
and QDOT redshift surveys of IRAS-selected galaxies. On large scales, this
anisotropy is caused by coherent peculiar motions, and gravitational
instability theory predicts a distortion of the power spectrum that 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. On
small scales, the distortion is dominated by the random velocity dispersion in
non-linear structures. We fit the observed anisotropy with an analytic model
that incorporates two parameters, beta, and a small-scale velocity dispersion
sigma_v. Tests on N-body simulations show that this model recovers beta quite
accurately on the scales accessible to the existing IRAS redshift surveys.
  Applying our procedure to the 1.2-Jy and QDOT surveys, we find beta=0.52 +/-
0.13 and beta=0.54 +/- 0.3, respectively. These results imply Omega
approximately 0.35 if galaxies trace mass, or a bias factor of about 2 if
Omega=1.
