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We analyzed ROSAT PSPC images of the bright, nearby $(z<0.1)$ galaxy clustersA401, A1656 (Coma), A2029, A2199, and A2256 to constrain their intrinsic shapesand baryon fractions. Following Buote & Tsai we probed the aggregate structureof the clusters on scales $\sim 1.5h^{-1}_{80}$ Mpc to reduce effects ofpossible substructure on smaller scales ($\lesssim$ a few hundred kpc). Theellipticities of the X-ray isophotes are typically $\epsilon\approx 0.15 -0.25$ and display negative radial gradients significant (except Coma). Byassuming hydrostatic equilibrium and a variety of mass models we obtainellipticities $\epsilon_{mass} \approx 0.40-0.55$ for isothermal models of thetotal gravitating matter of the clusters; the $\epsilon_{mass}$ constraintschange by $<10%$ upon consideration of the small temperature gradients shown byASCA to be typical for rich clusters; the observed X-ray ellipticity gradientsrequire that mass models with constant ellipticity have steep density profiles,$\rho\propto r^{-4}$. Estimates of the gas masses are highly insensitive to theellipticities of the X-ray isophotes. The clusters in our sample haveincreasing fractions of gas mass to total mass with radius and have$M_{gas}/M_{tot} = (4% - 11%)h^{-3/2}_{80}$ within a radius $1.5h^{-1}_{80}$Mpc, in excellent agreement with the results of White & Fabian and the BaryonCatastrophe proposed by White et al. (1993). Finally, the ellipticities of thedark matter distributions are essentially identical to $\epsilon_{mass}$ andare consistent with the shapes of dark halos predicted by N-body simulationsand the shapes of the galaxy isopleths in the clusters in contrast to ourprevious conclusions using {\it Einstein} data.Comment: 29 pages (2 figures), PostScript, to appear in ApJ on February 1,  1996. No changes to the text -- the publication date given before was  incorrec

X-Ray Constraints on the Intrinsic Shapes and Baryon Fractions of Five Abell Clusters