  We present analysis of the shape and radial mass distribution of the E4
galaxy NGC 3923 using archival X-ray data from the ROSAT PSPC and HRI. The
X-ray isophotes are significantly elongated with ellipticity e_x=0.15
(0.09-0.21) (90% confidence) for semi-major axis a\sim 10h^{-1}_70 kpc and have
position angles aligned with the optical isophotes within the estimated
uncertainties. Applying the Geometric Test for dark matter, which is
independent of the gas temperature profile, we find that the ellipticities of
the PSPC isophotes exceed those predicted if M propto L at a marginal
significance level of 85% (80%) for oblate (prolate) symmetry. Detailed
hydrostatic models of an isothermal gas yield ellipticities for the gravitating
matter, e_mass=0.35-0.66 (90% confidence), which exceed the intensity weighted
ellipticity of the R-band optical light, <e_R> = 0.30 (e_R^max=0.39).
  We conclude that mass density profiles with rho\sim r^{-2} are favored over
steeper profiles if the gas is essentially isothermal (which is suggested by
the PSPC spectrum) and the surface brightness in the central regions (r<~15")
is not modified substantially by a multi-phase cooling flow, magnetic fields,
or discrete sources. We argue that these effects are unlikely to be important
for NGC 3923. (The derived e_{mass} range is very insensitive to these issues.)
Our spatial analysis also indicates that the allowed contribution to the ROSAT
emission from a population of discrete sources with Sigma_x propto Sigma_R is
significantly less than that indicated by the hard spectral component measured
by ASCA.
