  Using a singular isothermal sphere model for the matter distribution of
foreground clusters of galaxies, we study the statistics of giant arcs in flat
cosmologies with and without a cosmological constant. We find that the relative
number of arcs predicted within $z=1$ in a universe with $\Omega_0=0.3$ and
$\lambda_0 \equiv \Lambda/(3H_0^2)=0.7$ is a factor of $\sim2$ larger than the
one in the Einstein-de Sitter universe ($\Omega_0=1, \lambda_0=0$). For a
luminosity-dependent evolution model of the number density of background
galaxies that accounts for the over-density of faint blue galaxies at
$z_s\approx0.4$, the Einstein- de Sitter cosmological model predicts that about
$5\%$ of clusters of galaxies with an X-ray luminosity $L_x > 2 \times 10^{44}$
erg s$^{-1}$ should have giant arcs with length-to-width ratio larger than 10.
This is a factor of $\sim4$ lower than the observed fraction in the
gravitational lensing survey of distant X-ray selected EMSS clusters of
galaxies, indicating that the matter distribution of clusters of galaxies
deviates significantly from simple isothermal spheres or/and the presence of a
significant cosmological constant. It is profitable to further study the
constraint on the cosmological constant from giant arcs using more realistic
cluster models.
