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No known class of extragalactic X-ray sources has been shown conclusively to have the properties necessary to explain the origin of the cosmic X-ray background (XRB). It has been suggested that star-forming galaxies could make a significant contribution to the XRB, and a convincing scenario for how these galaxies could have both high individual luminosities and the hard X-ray spectra required for such a contribution has been proposed (Griffiths \& Padovani 1990). We have employed the {\sl Einstein}, {\sl ROSAT}, and {\sl ASCA} observatories to examine this proposal.
Recently, a sample of 242 IR-selected {\sl ROSAT} sources was published and the discovery of a new class of X-ray--luminous normal spiral galaxies (20 objects) was claimed (Boller {\sl et al.} 1992). In addition to reexamining the classifications of 17 of these 20 objects, we have obtained optical spectra for 87 of the unclassified objects from the Boller {\sl et al.} sample and for 53 IR-selected objects from a new catalog of faint {\sl Einstein} sources. The majority of the objects observed are AGNs. While there appear to be a few H~II galaxies with high X-ray luminosities ($> 10^{42}$ ergs s$^{-1}$), only one of these, NGC 3256, is a confirmed X-ray source. An {\sl ASCA} observation of NGC 3256 indicates that its broad-band spectrum is very soft and that its true X-ray luminosity is at most $5 \times 10^{41}$ ergs s$^{-1}$ (.4 - 8 keV). Thus, there is no evidence for a class of X-ray--luminous H~II galaxies in these samples. Furthermore, the X-ray spectrum of NGC 3256 indicates that it and similar objects do not have the spectral characteristics necessary to explain the spectrum of the XRB. The most promising discovery in this survey is a substantial number of H~II/Seyfert composite galaxies. Their optical spectra are dominated by starburst characteristics; however, broad [O III] lines and sometimes very faint, broad H$\alpha$ emission betray their true Seyfert identities. Yet these optically innocuous galaxies have {\sl L}$_{\rm X}$ in the range $10^{42.7}-10^{43.7}$ ergs s$^{-1}$, typical for bright Seyfert 1 galaxies. The nature of these galaxies and their contribution to the XRB must be studied further.
