  (Shortened) We construct a subsample of the Lauer & Postman (LP) brightest
cluster galaxy (BCG) sample which comprises 64 Abell/ACO clusters with X-ray
luminosities from ROSAT. We find that both BCG metric luminosities and
residuals from the L-alpha relation of LP are significantly correlated with the
X-ray luminosity of the cluster at the 99.6% confidence level, in the sense
that more X-ray luminous clusters have brighter BCGs. The strength of this
correlation increases with increasing X-ray luminosity and with increasing
values of alpha. We obtain a new distance indicator for BCGs, the L-alpha-L_X
relation. Applying this new relation to our sample, we find that the frame
defined by these clusters has a bulk motion of 494 km/s in the CMB frame but
the 95% confidence range on the amplitude is 306 to 1419 km/s. These results
are inconsistent with this frame being at rest in the CMB frame at the 98.6%
confidence level. For the outer shell subsample (57 BCGs with cz > 6000 km/s),
the bulk motion is consistent both with no motion in the CMB frame and with the
motion found by LP. However, our analysis of the 107 Lauer and Postman BCGs in
this outer shell indicates that, even with no X-ray correction, the CMB-frame
bulk motion of these clusters is not statistically different from zero.
Furthermore, for the subsample with X-ray data, the X-ray correction goes in
the sense of reducing both the amplitude (by 663 km/s) and significance (from
98.8% to 83.8%) of its CMB-frame bulk motion, as well as reducing the internal
inconsistency between its motion and that of the remainder of the Lauer and
Postman sample with no X-ray data. Claims of large-scale, large-amplitude bulk
flows should be regarded with caution until X-ray data become available for
more clusters.
