The French Communist Party is going through a crisis of identity that is closely connected with its second-string position in President Francois Mitterrand's Socialist-led government.

After serving nearly two years as junior partners in the coalition, the Communists have gained a certain amount of the respectability that they sought. But they are losing their image as the natural party of the disgruntled and underprivileged. And they have failed to halt their long-term electoral decline.

Differences within the party over strategy were papered over at an important meeting of the party's policy-making Central Committee earlier this week, which ended with a solemn reaffirmation of Communist participation in the government. As is customary, the meeting "unanimously" adopted the report submitted by party leader Georges Marchais, despite earlier rumors that he was in danger of being replaced.

Recent public opinion polls have suggested that the Communist share of the vote is now down to a little over 10 percent from a high of 28 percent in the elections of 1946 when the party was the largest single political force in France. In the 1981 presidential election, the party received just over 15 percent in the first round.

This electoral decline has left the party uncertain about its place in French politics and its relationship with Moscow. Despite dabbling with Eurocommunism in the mid-1970s, the French Communists never went as far as the Italians or the Spanish in criticizing the Kremlin. At home, the party's attitude toward the Socialists has veered from cooperation to bitter rivalry and back again.

For their part, the Socialists seem to be attempting to kill the Communists with kindness. Socialist leaders constantly praise their Communist colleagues for their contribution to the government but make all the important decisions themselves.

By offering the Communists four relatively minor posts in his Cabinet, Mitterrand caused alarm bells to ring in Washington. His own view, however, was that the Communists were easier to control inside the government than out of it--particularly after he demonstrated that he did not need their votes for a majority in the National Assembly.

The slump in the Communist Party's fortunes has been particularly evident since 1972, when Marchais became its secretary general. During this period, the Communists have had to cede to the Socialists their claim to be the largest party of the left.

In a recent newspaper interview, sociologist Andre Touraine described the Communist Party as "a sand castle surrounded by the sea."

"The Communist Party doesn't have a future. From the moment when it ceased to be a Leninist party and was no longer identified with the whole of the left, which is now better represented by the Socialists, it was condemned to decline," he said. Touraine speculated that if the Communists left the government now, their share of the popular vote would fall quickly to beneath 10 percent. If they remained inside, the same decline would eventually occur but perhaps more slowly.

Here lies the heart of the Communists' dilemma. As allies of the Socialists, particularly during a period of gathering economic austerity, they have been forced to make one humiliating concession after another. But under the present electoral system, which favors the larger parties, they have no place else to go as long as they want to remain a significant political force.

They also have to accept the sometimes humiliating conditions imposed by the dominant Socialists. For the past few weeks, the noncommunist press has been describing with glee how the Communist Party is being made "to swallow snakes"--a French expression for taking a bitter pill.

Among the "snakes" the party has been made to digest are the mass expulsion of Soviet officials, the "Atlanticist" foreign policy under Mitterrand and a drastic austerity program at home that is likely to increase unemployment and cut into the average standard of living.

Symptomatic of the disquiet within Communist ranks was an internal document circulated by dissidents prior to the Central Committee meeting criticizing the government for not being sufficiently left-wing. The document said that the Socialists remained a "traditionally anti-Soviet" and "anticommunist" party.

Another attack on Marchais' leadership came in an open letter to the Central Committee from Jeannette Thorez-Vermeersch, the widow of the party's former leader Maurice Thorez and widely regarded as a representative of the pro-Moscow faction. She accused the leadership of being "opportunist" and of "abandoning its revolutionary goals."

Marchais defended the government's record, praising nationalizations and social reforms such as retirement at age 60. He mocked the press by saying that he would gladly swallow "snakes such as these morning, noon, and night."

Perhaps the Communist Party's strongest card in its relations with the Socialists is its control over the CGT union federation, the largest in France. To a limited extent, Communist leaders have been able to exploit Mitterrand's belief that their presence in the government is necessary to prevent strikes and other protests breaking out at a time of economic hardship.

The need to reassure the Communists was one of the reasons that Mitterrand decided against switching prime ministers during a government shakeup in March. The Communist Party felt that its interests were best looked after by the present prime minister, Pierre Mauroy, and launched a campaign in his favor.