  Imaging of the Sunyaev-Zel'dovich (SZ) effect in galaxy clusters combined
with cluster plasma x-ray diagnostics can measure the cosmic distance scale to
high redshift. Projecting the inverse-Compton scattering and x-ray emission
along the cluster line-of-sight introduces systematic errors in the Hubble
constant, H0, because the true shape of the cluster is not known. I present a
study of the systematic errors in the value of H0, as determined by the x-ray
and SZ properties of theoretical samples of triaxial isothermal ``beta'' model
clusters, caused by projection effects and observer orientation. I calculate
estimates for H0 for each cluster based on their large and small apparent
angular core radii and their arithmetic mean. I demonstrate that the estimates
for H0 for a sample of 25 clusters have 99.7% confidence intervals for the mean
estimated H0 analyzing the clusters using either their large or mean angular
core radius are within 14% of the ``true'' (assumed) value of H0 (and enclose
it), for a triaxial beta model cluster sample possessing a distribution of
apparent x-ray cluster ellipticities consistent with that of observed x-ray
clusters. This limit on the systematic error in H0 caused by cluster shape
assumes that each sample beta model cluster has fixed shape; deviations from
constant shape within the clusters may introduce additional uncertainty or bias
into this result.
