NEW YORK -- The other night, a child asked his mother about 6-year-old Elizabeth Steinberg. The girl's picture had been on television, her story in all the papers. At that moment she was tethered to machinery in a hospital, brain dead but still breathing until someone in authority could be found to pull the plug. Then her parents would be charged with her murder.

Elizabeth Steinberg was the alleged victim of child abuse. That much is clear. But that, really, is not what made her story so gripping to New York. She was not your ''ordinary'' abused child. She was white and middle class. Her adoptive parents -- living together but not married -- were named Steinberg and Nussbaum. Hedda Nussbaum had worked at Random House, where she edited children's books. Joel Steinberg is a lawyer -- maybe also a drug abuser. Drugs of all kinds were found in the couple's filthy Greenwich Village apartment.

The private agony of Elizabeth Steinberg turns out to have been not so private after all. Neighbors had heard screams coming from the apartment. Both the little girl and her mother were seen bruised, and apparently had been beaten. Twice the cops had been summoned to the Steinberg apartment. Twice they went away and no arrests were made. Child-welfare workers also checked. They, too, had received calls. On one occasion, they had examined little Elizabeth in the nude, checking for bruises. They found none. But anonymous tipsters persisted -- some calling a hot line. One caller was told that she had to have firsthand information about a beating before anything could be done.

So Elizabeth fell between the cracks. One day her father -- and maybe her mother too -- allegedly beat her almost to the point of death. Then, panicked, the mother summoned the police. Beaten herself, her bones broken and her face battered, she said Elizabeth was choking on food and had fallen off her roller skates. Drugs were found in the apartment -- drugs and Mitchell, a 16-month-old boy who had also been adopted. He was found tied to a chair. Now the boy is a ward of the court, and questions are being asked about how both adoptions were permitted.

Stories like this tell us something about ourselves. We lament the suicides of suburban teen-agers, but seem to accept the ghetto violence that claims the lives of so many kids -- suicides of a different kind. We are seized with fear about drugs, but turn our attention elsewhere when coke and crack become yet another ghetto killer. This will be the pattern, too, with AIDS if it remains limited to homosexuals, intravenous drug abusers and minority groups. Then the rest of the nation will move on to a new concern. As always, out of sight is out of mind.

As for child abuse and domestic violence, it is the same. Cops and social workers know the extent of it. Their daily fare is child abuse and violence -- nearly 67,000 cases last year and more than 100 deaths. When the victims are black or Hispanic, when they are dark and poor, little attention is paid. For these people, the system -- underfinanced, overwhelmed, staffed by exhausted social workers with 20 to 30 cases apiece -- fails all the time. The cracks yawn wider, neighbors care less. The ghetto night has too many screams to be answered.

As a nation we've come a long way since the innocent 1960s, when we thought poverty and all its ills could be eradicated with dollars from Washington. We now know our limits. Our pockets are not deep enough, our intellect not keen enough, our good intentions sapped by cynicism and reality. Poverty, like a malevolent virus, mutates before our eyes -- changes, adapts and mocks our cures. Once we made war on it but, having been deprived of an outright victory, we seem to have given up the effort. The shocking pathologies that may have killed Elizabeth Steinberg -- rage, drugs, mental instability -- are commonplace in the ghetto. It took a middle-class white couple to put them on page one.

At the dinner table where I sat, little Max asked his mother about death -- about what it means to be brain dead. Slowly, carefully, walking a mine field through the fragile psyche of a child, she explained. But there is no good explanation for how an entire city -- cops, welfare workers, teachers -- let Elizabeth Steinberg die. Nor is there a good explanation for how this sort of thing not only happens here and across the nation, but how we have come so uninterested with it all.

For a tragic moment, we all had something in common with little Elizabeth. In our own way, we're brain dead too.