Molecular gas in absorption and emission along the line of sight to W31C
  G10.62-0.38
  We used the ARO 12m antenna to observe emission from the J=1-0 lines of
carbon monoxide, \hcop\ and HNC and the J=2-1 line of CS toward and around the
continuum peak used for absorption studies and we compare them with CH, HNC,
C\p\ and other absorption spectra from PRISMAS. We develop a kinematic analysis
that allows a continuous description of the spectral properties and relates
them to viewing geometry in the Galaxy. As for CH, HF, C\p, \hcop\ and other
species observed in absorption, mm-wave emission in CO, \hcop, HNC and CS is
continuous over the full velocity range expected for material between the Sun
and W31 4.95 kpc away. CO emission is much stronger than average in the
Galactic molecular ring and the mean \HH\ density derived from CH, $4 \pccc
\la$ 2$<$n(\HH)$>$ $\la 10 \pccc$ at 4 $\la$ R $\la$ 6.4 kpc, is similarly
elevated. The CO-\HH\ conversion factor falls in a narrow range \XCO\ $=
1-2\times10^{20}~\HH\ \pcc~({\rm K}-\kms)^{-1}$ if the emitting gas is mostly
on the near side of the sub-central point, as we suggest. The brightnesses of
\hcop, HNC, and CS are comparable (0.83\%, 0.51\% and 1.1\% respectively
relative to CO) and have no variation in galactocentric radius with respect to
CO. Comparison of the profile-averaged \hcop\ emission brightness and optical
depth implies local densities n(H) $\approx 135\pm25\pccc$ with most of
excitation of \hcop\ from electrons. At such density, a consistent picture of
the \HH-bearing gas, accounting also for the CO emission, has a volume filling
factor 3\% and a 5 pc clump or cloud size.
