We use X-ray observations obtained with the Einstein Observatory to constrain the shape of the dark matter in the inner regions of Abell clusters A401, A426, A1656, A2029, and A2199, each of which exhibits highly flattened optical isopleths. We model the dark matter as an ellipsoid with a mass density rho approximately r-2. Making the usual assumptions that the X-ray gas is an ideal gas in hydrostatic equilibrium, we then construct model X-ray isophotes. Because we use the ratio of potential depth to gas temperature as a fitting parameter, our analysis is insensitive to the poorly known temperature of the gas or to the temperature gradient. We constrain the possible shapes of the dark matter by comparing these model isophotes to the image isophotes. The X-ray isophotes, and therefore the gravitational potentials, have ellipticities of approximately 0.1 - 0.2. On comparison of the derived dark matter shapes to the optical shapes given by Carter & Metcalfe, we find the dark matter within the central approximately 1 Mpc to be substantially rounder for all the clusters. Typically, the ellipticity of the galaxies is approximately 0.4 - 0.6, whereas that of the gravitating matter is constrained to be approximately 0.1 - 0.3. The position angles of the two distributions are consistent within the approximately 10-degrees - 20-degrees uncertainties. By limiting our study to the inner regions of the clusters, we believe we are relatively insensitive to effects of possible substructure. Therefore, we conclude that the shape of the galaxy distribution in these clusters traces neither the gravitational potential nor the gravitating matter.
