  We extend earlier work by studying in the quiet Sun the line profiles of the
hydrogen Lyman-alpha and Lyman-beta lines, which were obtained
quasi-simultaneously in a raster scan near disk center. The self-reversal
depths of the Ly-a and Ly-b profiles are quantified by measuring the maximum
spectral radiances of the two horns and the minimum spectral radiance of the
central reversal. The information on the asymmetries of the Ly-a and Ly-b
profiles is obtained through the calculation of the 1st and 3rd-order moments
of the line profiles. We find that the emissions of the Lyman lines tend to be
more strongly absorbed in the internetwork, as compared to those in the network
region. Almost all of the Ly-a profiles are self-reversed, whilst about 17% of
the Ly-b profiles are not reversed. The ratio of Ly-a and Ly-b intensities
seems to be independent of the magnetic field strength. Most Ly-a profiles are
stronger in the blue horn, while most Ly-b profiles are stronger in the red
horn. However, the opposite asymmetries of Ly-a and Ly-b are not pixel-to-pixel
correlated. We also confirm that when larger transition-region downflows are
present, the Ly-a and Ly-b profiles are more enhanced in the blue and red
horns, respectively. The first-order moment of Ly-b, which reflects the
combined effects of the profile asymmetry and motion of the emitting material,
strongly correlates with the Doppler shifts of the Si iii and O vi lines,
whilst for Ly-a this correlation is much weaker. Our analysis shows that both
Ly-a and Ly-b might be more redshifted if larger transition-region downflows
are present. We also find that the observed average Ly-b profile is redshifted
with respect to its rest position.
