Perhaps it was the decapitation of one young woman and the slasher slaying of another in an apartment complex that riveted attention on the growing frequency of murder in Houston.

Or it may have been the four children, aged 2 to 12, who were bound and left to die in a house fire set by their assailant.

Or maybe it wasn't any one act of brutality but rather the enormity of the collective carnage. Whatever it was, by the time the number of murders this year had reached 398, the Houston Chronicle declared, "Houston, known in the 1950s as the murder capital of the nation, may be on the way to regaining that dubious distinction."

The toll has since risen to 403, up 40 percent from last year, a frightening increase over the usual 10 to 12 percent rise from year to year. So common has violent death become here that the Chronicle bragged during one lull: "Houston," the headline said, "goes without a murder, first time in six days."

The 403 killings compare to 280 at the same time in 1978, which, with 461 homicides for the entire year, itself established a record. That difference means one person was killed, on average, every 19 hours last year and every 14 hours this year.

(By comparision, Washington murder rate is up 18 percent this year, but remains, with 136 killings so far, well below the rate in Houston.)

Houston's rate cannot be explained away by sheer population growth. "If we knew why," says homicide Lt. C. J. Lofland of the increase, "we could take some positive action to stop it."

So serious has the problem become that Mayor Jim McConn openly fears that this "tremendous increase in murders" could lead "major corporations and the things that have made Houston good by moving in here [to] take a second look at the city."

The police department, for its part, has called together a group of sociologists and others to explore the possible causes of the increase in homicides.

Compared with other cities, Houston's homicides have always included a higher proportion of killings related to robberies. But neither that, nor any other single element, appears to account for the increase in murders this year.

Lofland says, in fact, the only difference is that Hispanics seem to account for a greater share of the killings than before.

"Bedrooms and barrooms," former mayor and now Chamber of Commerce president Louis Welch says of the vast majority of the killings. "And the bedroom's even worse than the barroom.

"Now how do you protect against the bedroom murder?"

But some of the most shocking murders this year have been far removed from the bedroom and barroom quarrels that ended in violent death:

A 28-year-old magician from West Germany was shot to death during a downtown street robbery last month while he was here as part of one of Houston's most prized industries, the convention.

A 33-year-old secretary was sexually abused and killed in her high- rent southwest Houston apartment. She was decapitated with a butcher knife; police have yet to find her head.

Two weeks later, a 27-year-old woman was found in the same apartment complex, stabbed repeatedly, her lifeless body lying next to a butcher knife.

A Memorial-area couple and their 14-month-old son were shot and killed in their home, powder burns remaining on their heads from the .22 caliber long-rifle pistol that killed them.

Unlike Mayor McConn, Welch doesn't believe Houston's shiny image as a good place to do business has been threatened by the murders and is only concerned about "how much mileage" the out-of-town press can get out of it.

"There was an awful high murder rate in the winning of the West," he offered. "But nobody wants to give up Arizona and California on account of it."

In a sense, the West still is being won here.

"The commitment to weapons and the fantasy of the Wild West," says University of Houston sociologist William Simon, leads to an aggressive, "quasi-rural" type of crime in the nation's fifth largest city.

You can see it, he says, even in the way people drive -- "cutting in and out of lanes, macho driving, one hand on the steering wheel and one hand holding a beer and driving like they're going over the open range."

Perhaps not coincidentally, traffic deaths are also up this year, by 25 percent, from 176 to 218.

Wild driving and abundant homicide, Simon said, may both spring from what he sees as a major problem with law enforcement here, namely that the city is basically unwilling to tax itself to protect itself.

"Houston is underpoliced," Simon said, "and the police are underpaid, undertrained and underdisciplined. That creates a general lack of respect for the law, whether it's going through a red light or robbing a convenience store.

"The city doesn't take it very seriously, so the people don't take it very seriously."

Houston has about 3,000 police officers, about half of what is needed to police one of the most sprawling cities in America, according to a spokesman for McConn.

More uncertain is whether the economic boom that has sparked Houston for a decade also is feeding the rise in murder. "There is a kind of anger here," said Simon, "As the illusion of success outraces the reality."

That is particularly true for the blacks and Mexican-Americans whose lack of skills may make them more marginal in the thriving job market around them.

"To the extent immigration [to Houston] is a cause of the increase, it could then reflect the kinds of people moving here," says another University of Houston sociologist, Sol Tannebaum. And his guess, based on limited data, is that Hispanics may account for a disproportionate share of the surge in homicides.

Over the past 10 years, Hispanics' share of homicides have increased from less than 10 percent to more than 25 percent. Their actual proportion in the population is unknown, given the community of thousands of illegal aliens here.

"And my guess," Tannebaum adds, "is that it's the more recent residents rather than the older Spanish population."

All of this has been going on while the state of Texas is sending people to death row so fast it should soon surpass Florida as the state with the most people waiting for execution. One of the first states to have its revised death penalty law upheld by the Supreme Court, Texas now has 124 people on death row, according to the Texas Department of Corrections, up from 111 at the beginning of the year.

But the death penalty does not appear to be much of a deterrent to the "bedroom and barroom" murders that account for most of the slayings in Houston.

Texas law, Texas courts and Texas juries essentially sanction a lot of killing. "On the whole," says Tannebaum, "Texans apparently live in a system where homicide is an acceptable phenomenon. It's not liked, it's not wanted, but it's tolerated."

He cited one study of homicides here, "Murder in Space City," that found "you could literally draw a line -- the closer the victim and the offender are to each other, the less anything will happen" to the killer in court. And the more distant they are from each other, the more it is likely that something will happen."

And so last week, a 26-year-old woman here was sentenced to five years in prison for the killing of her husband. And last February a 29-year-old woman was acquitted of murdering her husband, despite her admission that she shot him and then sawed him into five pieces, which she then hauled to California.

It happened in Houston.