Rioting spread through many parts of London and more than a dozen other cities in England this weekend as large gangs of youths smashed windows and looted shops, overturned and burned cars, and attacked police with bricks, bottles and Molotov cocktails.

The disturbances Friday and Saturday nights were the most widespread of the urban violence that began nine days ago, affecting scattered areas of London, Birmingham. Manchester, Liverpool, Leicester, Nottingham, Sheffield, Wolverhampton and other industrial cities and towns. But intensive use of police officers in most places produced many more arrests and kept damage and injuries below what was suffered at the height of rioting in London, Liverpool and Manchester last week.

Merchants in many urban areas across the country, including some neighborhoods where there has not yet been rioting, closed for the entire weekend or operated Saturday out of boarded-up shops. Streets were heavily patrolled by police, whose weekend and vacation leaves were cancelled, and tension was kept high, even in peaceful inner-city neighborhoods, by rumors about where trouble would break out next.

As the rioting went into its second week, British politicans stepped up their debate over causes of the unrest and whether tougher law enforcement measures or steps to deal with wide-spread unemployment and inner-city deterioration are the best remedies. These questions pose a serious political problem for Prime Minister Margaret Thatcher. She was elected on promises to vigorously defend law and order and revitalize the battered British economy -- and now appears to be failing to accomplish either aim.

Thatcher and some Cabinet ministers have largely denied that rapidly rising unemployment, the government's greatest political liability, is a primary reason for the upheaval. But other Cabinet ministers are pushing for emergency government programs to provide more make-work job training for the young unemployed and to help deteriorating areas.

Thatcher has confined her public statements to the issue of law and order. Friday night, in a speech before British parliamentary journalists, she condemned the "terrorism and criminal looting and thuggery we have witnessed in the past week.

"The veneer of civilization is very thin," she added. "It has to be cherished if it is to continue. All of a sudden we realize we should not take it for granted any more. We may have to do something about it day by day."

Besides approving better protective gear for the police, her Cabinet is considering legislation to make it easier to arrest and convict rioters. This has been advocated by right-wing conservatives and newspapers but not requested by the police.

The government also is considering a plan from one of Thatcher's critics in the Cabinet, Employment Secretary James Prior, to spend enough money on job training to provided government help, if not a job, for everyone who leaves school by Christmas. Prior has been the only Cabinet member to say openly that widespread youth unemployment, particularly high in areas where disturbances have occurred, could be an underlying cause of the violence. w

But in a televised speech this week, Thatcher told Britons once again she could not abandon her government's monetarist economic course by trying "short-cut solutions" to cut the 11 percent unemployment rate. She said the "convenient cure-all, to get the government to spend and borrow and print large sums of money to expand the econony, sounds so caring and compassionate. I only wish it were as easy as that."

Just such a massive increase in government spending on job creation and investment in nationalized industries was advocated Saturday by the opposition Labor Party leader, Michael Foot. He told a mineworkers' convention that the "background" of the riots "is the return to mass unemployment and particularly unemployment for young people, on a scale that most of us believed had been banished from our country forever."

The worst weekend rioting was in London, where police reported they sustained 60 injuries and made more than 500 arrests battling white, black and Asian youths throwing rocks and Molotov cocktails and breaking into stores in a ring of crowded, lower-income, racially mixed neighborhoods encircling the city's affluent core. Buildings and overturned cars were burned and shops were looted, mostly by hit-and-run gangs chased by police.

The most damage occurred in the racially-mixed area of Brixton in south London, the scene of devastating rioting in mid-April that preceded the current wave of urban disturbances.

Saturday afternoon in London, several hundred Asian youths broke windows and threw missiles at police after a prayer service for a Pakistani woman and her three children who were killed in an arson attach on their home.

Asian immigrants blame the attack on white East London toughs called "skinheads" because of their close-cropped hair. Asian youths ran through Walthamstow in northeastern London shouting, "We want skinheads," but they were dispersed by hundreds of police who sealed off the area.

Police also battled gangs of youths Saturday afternoon in the city center of Birmingham, where violence broke out for the first time Friday night in racially mixed neighborhoods. Several buildings were set on fire and dozens of shops were looted by gangs of black and white youths both Friday and Saturday nights. Birmingham's central shopping center was shut for a time Saturday afternoon when several hundred youths, many of them skinheads, went on a rampage breaking windows.

Gangs of youths battled police in Nottingham and Hull both weekend nights, and there were disturbances in many other cities and towns in the nothern half of England, where the effects of Britain's economic decline and rapid rise in unemployment have been most severe.

"The lunatic fringe have seen it happening on television and wanted to prove they were men as well," said Nottinghamshire Chief Constable Charles McLachlan. "I hope they've now got it out of their system. It was not racial violence. Both black and white youths were making an attack on the police and authorities."

One focus of the political debate has been the campaign for next Thursday's by-election to fill an empty Parliament seat in the industrial city of Warrington, between Liverpool and Manchester. Attention was first focused on the campaign as a test of voter strength in a Labor Party stronghold for the new centrist Social Democratic Party, whose candidate is former Labor Cabinet member Roy Jenkins.

Since the rioting began in nearby cities and London, Warrington has become a stopover for nationally known politicians expounding conflicting theories about the riots while campaigning for their parties' candidates in the by-elections. Former Labor Cabinet member Shirley Williams, now a Social Democrat, went to Warrington to charge that radical leftists who have "climbed into the Labor Party" in recent years could be linked to the urban violence.

Home Secretary William Whitelaw also was in Warrington, campaigning for the Conservative candiate, when he announced that he has asked the country's chief constables for reports on the possible role of political agitators in the disturbances. He said police believed the rioting in nearby Manchester appeared organized at times, although he had no evidence of it.

"There are all sorts of different motives in different areas," he added. "There have always been agitators who latch on to the trouble wherever it is. That is the case in any community. I do not doubt that they have done it in this case."