A French experiment. Can France fly on left and right wing?

France's Socialist president, Fran,cois Mitterrand, has come through last week's parliamentary elections with a stronger political hand than expected. Although a conservative coalition gained by far the most votes, it won only the slimmest of majorities in the new parliament.

This left Mr. Mitterrand and his Socialist Party, still the largest single party in Parliament, in a better bargaining position than predicted. In particular, it gave the President more room for maneuver in selecting and working with a political rival as his new prime minister -- a first in the 28-year history of the Fifth Republic.

At press time it appeared that Jacques Chirac, the neo-Gaullist mayor of Paris and leader of the conservative coalition, would form the new government. It was unclear, however, just how this unprecedented power-sharing arrangement between a Socialist president and a right-wing prime minister would work.

Political instability or even early elections now loom on the horizon. In the past, the French president enjoyed a clear majority in the legislature, giving him much more freedom of action than an American president. The election means he must work with a conservative parliamentary majority. But at the same time that conservative coalition finds itself squeezed between an unexpectedly confident Mit-terrand and a suprisingly suc-cessful extreme-right National Front Party.

How this constitutional challenge is resolved will go a long way to determine the country's future shape.

After years of passionate antagonisms and violent swings, will France finally stabilize politically?

After building up its own nuclear force and asserting large influence in Europe and the third world, will it maintain its assertive role in the world?

And after a generation of dynamic social and economic transformation, will it be able to remain a tolerant society while keeping up with the Japan and the United States in the global economic race?

Messrs. Mitterrand and Chirac must first answer the political question. Pair of personal opposites

The two men are personal opposites. Mitterrand is the ultimate political tactician, a veteran of the shifting parlia-mentary alliances of the post-war Fourth Republic. His reserved, sphinx-like temperament and ability to broker for power earned him the nickname ``the Florentine,'' after the political schemers of Renaissance Italy.

Even his closest friends say that the President doesn't reveal his true feelings to them. Rather than addressing them with the informal ``tu,'' he prefers to use the formal ``vous.''

In contrast, Chirac makes blunt statements in public before assessing their political consequences. His agressive, energetic style has earned him the nickname of ``bulldozer.'' Rising before dawn during the campaign, he shook hands until well after midnight. At every stop, he made sure to join locals in a meal.

Despite their differences, pressure will be focused on the two men to find grounds for accord. Polls show that the public prefers Mitterrand to serve out the remaining two years of his term. The message seems to be that France no longer needs a monarchical, authoritariam style of leadership. Instead its right-left divide can be bridged by compromise. Essentials agreed upon

The relaxed atmosphere of the campaign underlined this preferance for a calm transfer of power. The essentials were agreed upon, namely that France should remain an independent nuclear power while supporting the Western Alliance and that it should make its economy more competitive in world markets without dismantling social benefits.

Mitterrand followed in the legacy of the Fifth Republic's founder, Charles de Gaulle, by modernizing the nuclear force de frappe and continuing to play a protectorship role in Chad. The formula helped reconcile France with her role in this superpower dominated world. Unlike either West Germany or Britain, torn by self-doubt and a feeling of dependence on the United States, Mitterrand's France felt confident enough to almost unanimously back the installation of American missiles in West Germany and take the lead in Europe in combating Russian espionage.

The new governing team must guard this confidence. That first means resolving the hostage crisis in Lebanon. Although both Mitterrand and Chirac agree on the basic outlines of French Mideast policy, it remains unclear whether they will be able to agree on the strategy to obtain the release of the seven kidnapped Frenchmen.

Beyond this, it remains unclear whether differences on such issues as the American Strategic Defense Initiative, which the opposition supports and Mitterrand opposes, will paralyze the government. Questions at home

Similar questions must be answered on the home front. In the 30 years between 1945 and 1975, the French economy expanded at a rate second only to that of Japan, faster than the United States and faster than West Germany. Most of this growth was not achieved by state-directed development. Although both Mitterrand's Socialists and Chirac's Gaullists now are convinced that more entrepreneural initiative is needed in the new high-tech world, they disagree over the details.

Chirac wants to proceed with a program of denationalizations, tax cuts, and public spending reductions while ending price controls, and making it easier for employers to fire workers.

Mitterrand might agree to some of these measures, but surely not to all of them. And the election seemed to say that the public wanted no radical changes. As a result, Chirac's program will have to be implemented much slower than intended, or not at all. Will tolerance continue?

A final unknown is whether France can remain a tolerant society. Economic growth produced a new urban society and brought 4 million immigrants from the third world. As crime and unemployment has risen in recent years, so has racism.

The Socialists and the conservatives both want to increase security while toughening nationality procedures. This may lead to a repressive program.

One sign was the emergence of the extreme-right National Front as a major political force. Led by the controversial Jean-Marie Le Pen, it won nearly 10 percent of the vote and is entering the National Assembly for the first time. Analysts credited the sucess to Le Pen's strident anti-immigrants, ultra-nationalist rhetoric.

Although mainstream conservatives reiterated after the election that they will not form an alliance with Le Pen, they might change their minds if their majority comes under attack.

At stake as France moves to answer these questions is whether the country can forge a new, more mature political system -- an orderly transfer of power, in democratic fashion, without destroying the existing institutions.

Before the Socialists came into power in 1981, the conservatives had ruled France for 23 consecutive years. The Socialists took the first step in proving that such a transfer of power was possible by completing their five-year legislative term without the predicted turbulence. The conservative victory last week means that France must now take the second step.