  Self-gravitating systems such as elliptical galaxies appear to have a
constant integrated specific entropy and obey a scaling law relating their
potential energy to their mass. These properties can be interpreted as due to
the physical processes involved in the formation and evolution of these
structures. Dark matter halos obtained through numerical simulations have also
been found to obey a scaling law relating their potential energy to their mass
with the same slope as for ellipticals, and very close to the expected value
predicted by theory. Since the X-ray gas in clusters is weakly dissipative, we
test here the hypothesis that it verifies similar properties. Comparable
properties for the dark matter component are also investigated. With this aim,
we have analyzed ROSAT-PSPC images of 24 clusters, and fit a S\'ersic law to
their X-ray surface brightness profiles. We found that: 1) the S\'ersic law
parameters (intensity, shape and scale) describing the X-ray gas emission are
correlated two by two, with a strong correlation between the shape and scale
parameters; 2) the hot gas in all these clusters roughly has the same
integrated specific entropy, although a second order correlation between this
integrated specific entropy and both the gas mass and the dynamical mass is
observed; 3) a scaling law links the cluster potential energy to its total
mass, with the same slope as that derived for elliptical galaxies and for dark
matter halo simulations. Comparable relations are obtained for the dark matter
component. All these correlations are probably the consequence of the formation
and evolution processes undergone by clusters of galaxies.
