  We describe and test a method to quantitatively classify clusters of galaxies
according to their projected morphologies. This method will be subsequently
used to place constraints on cosmological parameters ($\Omega$ and the power
spectrum of primordial fluctuations on scales at or slightly smaller than that
of clusters) and to test theories of cluster formation. We specifically address
structure that is easily discernible in projection and dynamically important to
the cluster. The method is derived from the two-dimensional multipole expansion
of the projected gravitational potential and yields dimensionless {\it power
ratios} as morphological statistics. If the projected mass profile is used to
characterize the cluster morphology, the power ratios are directly related to
the cluster potential. However, since detailed mass profiles currently exist
for only a few clusters, we use the X-ray--emitting gas as an alternative
tracer of cluster morphology. In this case, the relation of the power ratios to
the potential is qualitatively preserved. We demonstrate the feasibility of the
method by analyzing simulated observations of simple models of X-ray clusters
using the instrument parameters of the ROSAT PSPC. For illustrative purposes,
we apply the method to ROSAT PSPC images of A85, A514, A1750, and A2029. These
clusters, which differ substantially in their X-ray morphologies, are easily
distinguished by their respective power ratios. We discuss the suitability of
this method to address the connection between cluster morphology and cosmology
and to assess whether an individual cluster is sufficiently relaxed for
analysis of its intrinsic shape using hydrostatic methods. Approximately 50
X-ray observations of Abell clusters with the PSPC will be amenable to
morphological analysis using the method of this paper.
