It was a parking ticket, a fluke, that finally led police to "Son of Sam" after he had gunned down seven young women. A pedestrian recalled seeing a man running for a car along a street just visited by a "meter maid" issuing parking citations. Police reviewed the tickets and found only one given to a non-neighbored resident: David Berkowitz of Yonkers.

A $3,000 reward cracked San Francisco's frightening "Zebra" slaings, piquing the greed of an informant who figured the "Death Angels" behind the random killings of 12 whites in a series of unprovoked attacks.

In Chicago, it was a curious beat cop's missing persons report that first linked John Wayne Gacy, subcontractor and convicted sex offender, to one of 33 boys he'd strangled and buried under his house. He had lured them home with promises of drug parties and well-paying jobs.

Every cracked murder case, no matter how perplexing, has its "lucky break" -- that one magic piece of information that brings the puzzle together. Baffled Atlanta police are still hunting for their piece, as they try to track down the mysterious killer or killers who have snatched 21 black children off litter-strewn streets of the city's impoverished South Side over the last 19 months without leaving a trace.

Veteran investigators from other well-known murder cases say the break will likely come along. But will Atlanta police be able to recognize it? That, say the veterans, depends on how much old-fashioned detective work is done, how familiar investigators become with ever fact and nuance of each case -- no easy task because of the large number of victims.

Ninteen children have been found murdered here; two remain missing. All but two of the victims were boys, between seven and 15. Nine were strangled or asphyxiated; others were shot, bludgeoned or stabbed to death. Only scantly physical evidence -- similar fibers from possibly a carpet or a rope -- links as many as six of the victims, mostly street-smart children who were products of poverty and broken homes.

"This kind of killer has a whole different type of persona and is difficult to detect," speculates Dr. Richard G. Rappaport, a Chicago forensic psychiatrist who interviewed Gacy for the defense. "Later, police will see characteristics about the guy they should have known or suspected but likely won't know until after they find him."

Some officials theorize it could be only a male "authority figure" -- a preacher, a bus driver, a letter carrier, a policeman or an ex-policeman -- who could be in a position to spirit the children away undetected, without a struggle. Indeed, Gacy sometimes tried to pass himself off as a policeman.

In Atlanta, an army of police, including a 35-person special task force, scores of FBI agents and county lawmen, have pulled out the stops: roadblocks, lie detector tests, maps and charts that match up the victim's final hours and friendships, helicopters, tracking dogs, psychics, citizen tipsters, a computer to collate leads, forensic pathologists armed with microscopes, psychiatrists who draw profiles of prospective killers. A $100,000 reward goes begging.

Even former governor Lester Maddox, in a newspaper ad, pleaded with the killer to surrender to him, pledging $10,000 to his defense fund, and Eddie Garland, a well-known criminal lawyer, has promised to defend him if he will just turn himself in. Nothing has worked.

In pursuit of one popular theory -- that the killer is a John Gacy type, a psychopath with an appetite for sadism who is able to blend into the community by day in a respectable job, preying on his victims at will -- police are investigating the seamiest side of Atlanta's homosexual subculture, sources say, using young informants.

One task force investigator is assigned to each case, with the responsibility of becoming so familiar with his victim -- as well as the other victims -- that he will instantly recognize the magic piece to the puzzle that police figure will eventually lead them to the killer. Beat patrolmen are supposed to be briefed daily, but much case data has apparently trickled down slowly.

Last week, for example, Atlanta police officers knocked on the door of Patrick Rogers to inquire if his mother had seen him -- he was wanted on a burgulary warrant, they said. Not only has Patrick been dead for two months, his shocked sister told police, but he was one of the murdered black children, found in a river in December.

Such incidents give retired officers like W. K. Perry, a former Atlanta homicide chief, and others, including some task force investigators, plenty of ammunition to criticize Atlanta police. Plagued by a recent task force shakeup, police have been sniped at for their general bafflement, jurisdictional squabbles, mishandling of evidence, the death of homicide experience recruited for the task force initially and the one-year wait to start a full-scale investigation after the first body was found.

But Dan Sweat, president of Central Atlanta Progress, an association of downtown business boosters, says it is "unfair to judge them so quickly." He cites various murder cases that were slow to unravel, especially the "Yorkshire Ripper" case in England, which took five years to solve at a cost of $10 million. A policewoman busted open that case by suggesting the Ripper squad interrogate a truck driver detained for a license plate theft. He soon confessed to hammering and butchering 12 of 13 women believed to have been Ripper victims.

"It was a lucky break," said Assistant Chief Constable James Hobson, who experienced a fusillade of public criticism while working on the case. "But you have to be ready to recognize your lucky breaks. You make your own luck. You need a lot of good hard detective work, then the lucky breaks will come."

Atlanta police thought they had one last week when a 24-year-old white man was found dead, a suicide, in a light green Dodge. They'd been on the lookout for a white man in a green car as a possible witness to circumstances surrounding the murder of one black child.

The mayor rushed to the scene, grim-faced and "no commenting." Indeed, police said the man bore a strong resemblance to the composite drawing given by a witness under hypnosis. Some observers were even speculating the man was the suspect who had finally turned his rage from young boys on himself. Psychiatrists say such a scenario is not unlikely with psychopaths who murder. As reporters were broadcasting the body's resemblance to the sought-after witness, police finally escorted the woman who provided the composite to the downtown morgue to see the victim. Not even close, she said.

"If we could break it from sheer police work, we would have broken it by now," says Fulton County District Attorney Lewis Slaton. "People can't be dumping bodies without somebody seeing something. It's possible we've got the information right in front of us and don't know it."

That was the concern of Joseph Borelli, 49, the New York City homicide inspector who headed 300 investigators in the "Son of Sam" case. To prepare his men to be able to seize the opportunity from a parking ticket, he divided them into two groups. One worked the angle that each case had a different killer; the other group assumed every case was connected.

Until the ticket, the evidence consisted of spent bullet fragments from a .44 magnum revolver, which suggested one gunman. Policer visited gun stores in the area that carried the ammunition; they checked gun purchase records and registration. Nothing. Berkowitz, police discovered after his arrest, had bought the gun from someone who purchased it in Texas.

"When the break comes," says Borelli, "it's usually at the expense of another tragedy. Each time an incident of violence occurs, the killer allows the possibility that he will make a mistake. You just have to be ready for it."

Some local authorities and psychiatrists say the increasing rate of disappearances of children here since January suggests the killer may be getting bolder and losing control of his compulsion, which could lead to a mistake and his capture.

The last four children have disappeared on an average of every two weeks, starting with 14-year-old Lubie Geter on Jan. 3. Three of the children have been found murdered. The fourth, Curtis Walker, 12, has been missing since Feb. 19. Before January, no child disappeared for two months. Before that, children vanished about every 3 1/2 weeks.

"Rarely do you get this number of victims," says FBI criminologist Robert Ressler, an expert on abnormal criminal personalities. "Gacy succeeded -- if you want to call it that -- because he picked up transients, people who wouldn't be missed. He was able to put up a good front, different faces, to manage his secret."

Atlanta's phantom killer "may have a business, social, even a family life," says Ressler. "The reward money and anything else is no good unless he shares it and people like this don't usually confide in anyone about their criminal activity. The killer is melting into the environment. No white guy in a business suit it putting kids in a briefcase. He is not taking white middle-class kids because he couldn't move in that society without being noticed."

Jack the Ripper was never brought to justice. Suddenly the murders stopped. Many theorize he killed himself. "The same thing could happen in this case," says one FBI agent. "He may become so consumed with his own rage and have no way to release it that he commits suicide and in the end we'll never know. The children's cases may never be solved."