Context : The sightline to W31C G10.62-0.38 was extensively observed in absorption under the PRISMAS program on Herschel .

Aims : To relate absorbing material to the older view of Galactic molecules gained from CO emission .

Methods : We used the ARO 12m antenna to observe emission from the J=1-0 lines of carbon monoxide , \mathrm { HCO ^ { + } } 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 ^ { + } 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 .

Results : As for CH , HF , C ^ { + } , \mathrm { HCO ^ { + } } and other species observed in absorption , mm-wave emission in CO , \mathrm { HCO ^ { + } } , 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 \mathrm { H _ { 2 } } density derived from CH , 4 ~ { } { cm } ^ { -3 } \la 2 < n ( \mathrm { H _ { 2 } } ) > \la 10 ~ { } { cm } ^ { -3 } at 4 \la R \la 6.4 kpc , is similarly elevated .
The CO- \mathrm { H _ { 2 } } conversion factor falls in a narrow range X _ { \mathrm { CO } } = 1 - 2 \times 10 ^ { 20 } ~ { } \mathrm { H _ { 2 } } ~ { } { cm } ^ { -2 } ~ { } ( { K } - \mbox { km s$ { % } ^ { -1 } $ } ) ^ { -1 } if the emitting gas is mostly on the near side of the sub-central point , as we suggest .
The brightnesses of \mathrm { HCO ^ { + } } , 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 \mathrm { HCO ^ { + } } emission brightness and optical depth implies local densities n ( H ) \approx 135 \pm 25 ~ { } { cm } ^ { -3 } with most of excitation of \mathrm { HCO ^ { + } } from electrons .
At such density , a consistent picture of the \mathrm { H _ { 2 } } -bearing gas , accounting also for the CO emission , has a volume filling factor 3 % and a 5 pc clump or cloud size .

Conclusions :