We describe (1) a new test for dark matter and alternate theories of gravitation based on the relative geometries of the X-ray and optical surface brightness distributions and an assumed form for the potential of the optical light, (2) a technique to measure the shapes of the total gravitating matter and dark matter of an ellipsoidal system which is insensitive to the precise value of the temperature of the gas and to modest temperature gradients, and (3) a new method to determine the ratio of dark mass to stellar mass that is dependent on the functional forms for the visible star, gas and dark matter distributions, but independent of the distance to the galaxy or the gas temperature. We apply these techniques to X-ray data from the ROSAT Position Sensitive Proportional Counter (PSPC) of the optically flattened elliptical galaxy NGC 720; the optical isophotes have ellipticity epsilon approximately 0.40 extending out to approximately 120'' (10'' approximately 1 kpc assuming a distance of 20h80 Mpc). The X-ray isophotes are significantly elongated, epsilon = 0.20-0.30 (90% confidence) for semimajor axis a approximately 100''. The major axes of the optical and X-ray isophotes are misaligned by approximately 30-degrees +/- 15-degrees (90% confidence). Spectral analysis of the X-ray data reveals no evidence of temperature gradients or anisotropies and demonstrates that a single-temperature plasma (T approximately 0.6 keV) having subsolar heavy element abundances and a two-temperature model having solar abundances describe the spectrum equally well. Considering only the relative geometries of the X-ray and optical surface brightness distributions and an assumed functional form for the potential of the optical light, we conclude that matter distributed like the optical light cannot produce the observed ellipticities of the X-ray isophotes, independent of the gas pressure, the gas temperature, and the value of the stellar mass; this comparison assumes a state of quasi-hydrostatic equilibrium so that the three-dimensional surfaces of the gas emissivity trace the three-dimensional isopotential surfaces-we discuss the viability of this assumption for NGC 720. Milgrom's Modification of Newtonian Dynamics (MOND) cannot dispel this manifestation of dark matter. Hence, geometrical considerations require, without mention of pressure or temperature, the presence of an extended, massive dark matter halo in NGC 720. Employing essentially the technique of Buote & Canizares (1992; Buote 1992) we use the shape of the X-ray surface brightness to constrain the shape of the total gravitating matter. The total matter is modeled as either an oblate or prolate spheroid of constant shape and orientation having either a Ferrers (rho is similar to r(-n)) or Hernquist density. Assuming the X-ray gas is in hydrostatic equilibrium, we construct a model X-ray gas distribution for various temperature profiles; i.e., isothermal, linear, and polytropic. We determine the ellipticity of the total gravitating matter to be epsilon approximately 0.50-0.70. Using the single-temperature model we estimate a total mass approximately (0.41-1.4) x 10(12)h80 M . interior to the ellipsoid of semimajor axis 43.6h80 kpc. Ferrers densities as steep as r-3 do not fit the data, but the r-2 and Hernquist models yield excellent fits. We estimate the mass distributions of the stars and the gas and fit the dark matter directly. For a given gas equation of state and functional forms for the visible stars, gas, and dark matter, these models yield a distance-independent and temperature-independent measurement of the ratio of dark mass to stellar mass M(DM)/M(stars). We estimate a minimum M(DM)/M(stars) greater-than-or-equal-to 4 which corresponds to a total mass slightly greater than that derived from the single-temperature models for distance D = 20h80 Mpc.
