  We report on 40 kiloseconds of ROSAT HRI observations of the globular cluster
M92 (NGC 6341) and 20 kiloseconds of observations of the globular cluster M13
(NGC 6205). We find that the low-luminosity (10^{32.5} erg/sec at 7.5 kpc)
source previously observed near the core of M92 with the ROSAT PSPC remains
unresolved at HRI resolution; we can identify it with M92 with 99 per cent
confidence. In M13 we find that the source seen with the ROSAT PSPC lies within
the core and is possibly associated with the cluster (96 per cent confidence).
We find probabilities of 99.8 per cent (M92) and 98 per cent (M13) for the
presence of additional unresolved emission within these globular clusters. We
interpret these results in light of current theories regarding the
low-luminosity sources.
