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We present the analysis and interpretation of a sample of eight ASCA observations of broad absorption line quasi-stellar objects (BALQSOs). This is the first moderate-sized sample of sensitive BALQSO observations above 2 keV, and the BALQSOs in our sample are among the optically brightest known ( B = 14.5-18.5). Despite the ability of 2-10 keV X-rays to penetrate large column densities, we find BALQSOs to be extremely weak sources above 2 keV, and we are only able to add two new 2-10 keV detections (0226-104 and IRAS 07598+6508) to those previously reported. By comparison with non-BALQSOs of similar optical continuum magnitudes, we derive the column densities needed to suppress the expected X-ray fluxes of our BALQSOs. In several cases we derive column densities ≳5 × 10 23 cm -2 for a neutral absorber with solar abundances. These are the largest X-ray column densities yet inferred for BALQSOs, and they exceed ROSAT lower limits by about an order of magnitude. Optical brightness does not appear to be a good predictor of 2-10 keV brightness for BALQSOs, but our data do suggest that the BALQSOs with high optical continuum polarizations may be the X-ray-brighter members of the class. For example, the highly polarized object PHL 5200 appears to be unusually X-ray bright for a BALQSO given its optical magnitude. We discuss the implications of our results for future observations with AXAF and XMM . If the objects in our sample are representative of the BALQSO population, precision X-ray spectroscopy of most BALQSOs will unfortunately prove difficult in the near future.
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