  We compare the observational and theoretical spectra of the $\Delta v$ = 2 CO
bands in a range of M dwarfs. We investigate the dependence of theoretical
spectra on effective temperatures as well as carbon abundance. In general we
find that the synthetic CO bands fit the observed data extremely well and are
excellent diagnostics. In particular the synthetic spectra reasonably match
observations and the best fit temperatures are similar to those found by
empirical methods. We also examine the \CDC isotopic ratio. We find that
fundamental $^{13}$CO bands around 2.345 and 2.375 $\mu$m are good
discriminators for the \CDC ratio in M dwarfs. The 2.375 $\mu$m is more useful
because it doesn't suffer such serious contamination by water vapour
transitions. Our current dataset does not quite have the wavelength coverage to
perform a reliable determination of the \CDC ratio in M dwarfs. For this we
recommend observing the region 2.31--2.40 $\mu$m at a resolution of better than
1000. Alternatively the observational problems of contamination by water vapour
at 2.345 $\mu$m maybe solved by observing at resolutions of around 50000. We
also investigated the possibility of using the $\Delta v$ = 1 CO bands around
4.5 $\mu$m. We find that the contamination due to water vapour is even more of
a problem at these wavelengths.
