  Galaxy redshift surveys provide a distorted picture of the universe due to
the non-Hubble component of galaxy motions. By measuring such distortions in
the linear regime one can constrain the quantity $\beta = \Omega^{0.6}/b$ where
$\Omega$ is the cosmological density parameter and $b$ is the (linear) bias
factor for optically-selected galaxies. In this paper we apply two techniques
for estimating $\beta$ from the Stromlo-APM redshift survey --- (1) measuring
the anisotropy of the redshift space correlation function in spherical
harmonics and (2) comparing the amplitude of the direction-averaged redshift
space correlation function to the real space correlation function. We test the
validity of these techniques, particularly whether the assumption of linear
theory is justified, using two sets of large $N$-body simulations. We find that
the first technique is affected by non-linearities on scales up to $\sim 30
\hMpc$. The second technique is less sensitive to non-linear effects and so is
more useful for existing redshift surveys. The Stromlo-APM survey data favours
a low value for $\beta$, with $\beta \simlt 0.6$. A bias parameter $b \approx
2$ is thus required if $\Omega \equiv 1$. However, higher-order correlations
measured from the APM galaxy survey (Gazta\~{n}aga and Frieman 1994) indicate a
low value for the bias parameter $b \approx 1$, requiring that $\Omega \simlt
0.5$. We also measure the relative bias for samples of galaxies of various
luminosity and morphological type and find that low-luminosity galaxies are
roughly three times less biased than $L^*$ galaxies. For the galaxy population
as a whole, we measure a real space variance of galaxy counts in $8 \hMpc$
spheres of $(\sigma^2_8)_g = 0.89 \pm 0.05$.
