Mrs. Thatcher's test: to win fast but not heavy-handedly

So far, so good.

But immense problems lie ahead, and harsh diplomatic and political pressures are building up on the British government here at home.

That was the situation facing Britain two days after it had landed what was said to be 5,000 men on the East Falkland Island. It was preparing to move quickly both south and east to attack the Port Darwin-Goose Green area, and Port Stanley itself.

The real problems may be just beginning.

On the military side, the Argentine Air Force is expected to launch new attacks on the men ashore. Defense experts warned that Argentina's three small, 1,800-ton submarines remained at large, each capable of sinking British ships.

Britain's aircraft were outnumbered and it had only two aircraft carriers. The loss of even one of them -- especially the Hermes -- would be a grievous blow to the war effort. Against this, British troops are professionals facing Argentine conscripts. British weapons are first-class and morale is high after recent successes.

On the diplomatic side, Britain is anxious to do two things that may turn out to be contradictory. It wants to complete the invasion as soon as it can, but it also needs to keep its own and Argentine losses to a minimum to retain as much support as possible and to bolster its own position of fighting to uphold the principles of democracy and international law.

London expects to see Common Market trade sanctions against Argentina renewed in Brussels May 24, if only for a short time. At this writing, the visit of the Pope to Britain also seemed likely to go ahead, since the decision lay with British bishops who were eager for the visit to take place.

Really heavy Argentine losses could worsen the British image and tarnish what it is fighting for. It could erode support for Britain at the United Nations and help create a solid anti-Western bloc of nations in Latin America and elsewhere.

Heavy British losses -- apparently avoided so far -- or a prolonged stalemate could lead to a loss of public support for the Thatcher government here at home.

The latest opinion poll, by Opinion Research May 23, showed 53 percent of just over 1,000 people who were questioned two days earlier ready to accept losing large numbers of British lives. Thirty-nine percent were opposed and 8 percent said they did not know.

The contradictory political pressures took this form:

If the government loses many men in taking Port Stanley, a number of Conservatives and military men in Britain will demand that only complete freedom for the islanders themselves can justify the losses. Yet that could mean defending the islands against a hostile and vengeful Argentina for a long time to come.

On the other hand, even the losses so far make many on the political left even more adamant in calling for a unilateral cease-fire and extended negotiations at the UN.

The actual landings were reported to have gone well. Britain said it lost a frigate (the Ardent), a Harrier jet, two small helicopters, and 48 men dead. British forces were said to have raised their flag in Port San Carlos and Ajax Bay, set up antiaircraft Rapier missiles and, according to official sources, shot down 20 Argentine aircraft (nine Mirages, five Skyhawks, two small Pucaras, and four helicopters).

On May 23 British forces ran aground the Falkland Island ferry M.V. Monsunen, which had been commandeered by Argentine forces after the April 2 invasion. The same day a bombing raid on the Goose Green airfield was said to have damaged a number of Argentine Pucaras.

The same day it was reported that two Argentine jets attacked British ships off the San Carlos bridgehead, and that one Skyhawk was shot down, bringing the total shot down since the British landing to 21.

Britain could now either tackle Port Darwin first and establish control over the grass airfield at Goose Green or do so while simultaneously aiming at Port Stanley.

The distance from Port San Carlos to Port Stanley is about 50 miles over marshy ground or bad roads. Helicopters were likely to be used. Troops would dig in in the hills above the capital and go in under cover of naval bombardment and air support.

For the moment, prominent politicians such as Tory defense specialist Julian Amery and Labour spokesman Peter Shore agreed May 23 that Mrs. Thatcher should move quickly to take Port Stanley before British supplies and hard-worked military equipment began to fail. They also wanted action before world opinion could try for a cease-fire disadvantageous to Britain.

Most British people seemed united in the conviction that Britain was doing something for the world. It was upholding a set of principles: that aggression cannot succeed, that the rule of law cannot be broken by a dictatorship or by anyone else, and that sovereignty cannot be ended by force of arms.

Over the horizon, somber questions lay in wait.

After paying tribute to Mrs. Thatcher's leadership qualities, the Sunday Times political editor, Hugo Young, posed some of them:

''Why should Britain's old-fashioned gesture in the Falklands persuade an aggressor anywhere else in the world to stay his hand? Why, even in Britain, shouldn't the moral be quite different: that hanging onto imperial anamolies requires an expenditure of men and money which is quite inordinate.''

The Falklands, he wrote, was a comparatively straightforward problem, much less complex than the Middle East or the arms race: ''If they (political leaders) cannot solve a small argument without war, what price a big one?''