  We present initial results of a program to obtain and analyze HST WFPC2
images of galaxies identified in an imaging and spectroscopic survey of faint
galaxies in fields of HST spectroscopic target QSOs. We measure properties of
87 galaxies, of which 33 are associated with corresponding \lya absorption
systems and 24 do not produce corresponding \lya absorption lines to within
sensitive upper limits. Considering only galaxy and absorber pairs that are
likely to be physically associated and excluding galaxy and absorber pairs
within 3000 \kms of the background QSOs leaves 26 galaxy and absorber pairs and
seven galaxies that do not produce corresponding \lya absorption lines to
within sensitive upper limits. Redshifts of the galaxy and absorber pairs range
from 0.0750 to 0.8912 with a median of 0.3718, and impact parameter separations
of the galaxy and absorber pairs range from 12.4 to $157.4 h^{-1}$ kpc with a
median of $62.4 h^{-1}$ kpc. The primary result of the analysis is that the
amount of gas encountered along the line of sight depends on the galaxy impact
parameter and B-band luminosity but does not depend strongly on the galaxy
average surface brightness, disk-to-bulge ratio, or redshift. This result
confirms and improves upon the anti-correlation between \lya absorption
equivalent width and galaxy impact parameter found previously by Lanzetta et
al. (1995). There is no evidence that galaxy interactions play an important
role in distributing tenuous gas around galaxies in most cases. Galaxies might
account for all \lya absorption systems with $W > 0.3$ \AA, but this depends on
the unknown luminosity function and gaseous cross sections of low-luminosity
galaxies as well as on the uncertainties of the observed number density of \lya
absorption systems.
