  (ABRIDGED)The Ly$\alpha$ clouds at low redshifts (z < 1.7)observed by HST
showed us two unexpected features: Lanzetta et.al.establised the association
between Ly$\alpha$ clouds and galaxies, and Ulmer pointed out the strong
clustering of Ly$\alpha$ clouds.
  Motivated by these observations together with another observation which
reports the detection of metal in the Ly$\alpha$ clouds at high redshift
universe, we propose the two-component protogalaxy modelas a model for the
Ly$\alpha$ cloud based upon the previous work.
  In our model,the Ly$\alpha$ clouds are supposed to be stable cold clouds
confined by the pressure of ambient hot gas in galactic halo.
  We determine the properties of these cold clouds and hot gas on the basis of
theoretical and observational constraints. Especially, we take into account the
stability of a cold cloud in the galactic halo in addition to the general
stability conditions in a two-component medium, and compare the derived
quantities of Ly$\alpha$ clouds both cases in galactic halo and in
intergalactic medium between high and low redshifts.
  We conclude that the condition that a cloud is stable against both
evaporation and tidal destruction by a hot galactic halo is very restrictive.
  As a result, in the most noteworthy case, at z${\sim}$0.5,it is concluded
that a pressure-confined, stable spherical Ly$\alpha$ cloud with typical
neutral hydrogen column density $N_{HI}=10^{14}cm^{-2}$ cannot survive in the
galactic halo, although much higher column density clouds of $N_{HI}=
10^{17}cm^{-2}$ can do there.
