  We present an analysis of the evolution of the Ly alpha absorption lines in
the redshift interval 0.0 to \sim 1.5 based upon a sample of 987 lines
identified in the spectra of 63 QSOs obtained with the Faint Object
Spectrograph on the Hubble Space Telescope. Fits to the number of absorbers per
unit redshift dN/dz = A \times (1+z)^\gamma yield values of \gamma in the range
0.1 to 0.3, decidedly flatter than results from groundbased data pertaining to
the redshift range z > 1.7. The combination of the space and groundbased data
suggest a marked transition in the rate of evolution of the Ly alpha lines at a
redshift of about 1.7. The distribution in redshift is well represented by a
power law and the distribution of equivalent widths above a rest equivalent
width of 0.1 \AA is fit well by an exponential. The sample of Ly alpha systems
which has ions from heavy elements has a much larger value of \gamma than a
sample of lines of comparable strength without heavy elements. We argue that
this is not likely due to either line misidentification or incomplete spectral
coverage and suggest that it is due to rapid evolution of the very highest
column density systems, rather than differences in metallicity. We also find
evidence that the rate of evolution increases with increasing equivalent width.
We compare our results for the variation of line density with redshift to
recent numerical simulations of Ly alpha absorbers, and find fairly good
agreement. Our results for the stronger Ly alpha lines in our sample are
compatible with the recent analysis of the absorber--galaxy correlation by Chen
et al. 1998 but for the weaker lines our results suggest that the association
between absorbers and galaxies is different from that for the stronger lines.
