density and CO column density per unit line width both increase monotonically with increasing excitation temperature. The solutions we have chosen are shown in Table \[bestLVG\_table\] and in Figure \[bestLVG\_figure\]. We emphasize that these solutions are not unique, but depend on our choice of $R$. However, the general behavior of the solutions are physically plausible, given that we expect the excitation temperature to increase as one moves from the cloud interior to the cloud periphery. This suggests that binning by $T_{ex}$ is a useful approach, and gives us a reasonable handle on how the physical conditions vary as a function of excitation temperature and position in the cloud.

Our assumption of 15 K for the kinetic temperature is a potential source of error in determining the carbon monoxide column density. To assess this, we have carried out some calculations using a kinetic temperature of 25 K which seems an upper limit to what one might expect in a cloud edge in a region with modest UV intensity. We find that for this value of the kinetic temperature, the column density per unit velocity gradient is approximately a factor 1.5 larger than for a kinetic temperature of 15 K, and the 