Gorbachev needs proof that risk-taking pays off. Superpower leadership on the line at summit. Reagan and Gorbachev approach their talks politically weakened and struggling with their nations' economies. Both are hoping for achievements that will restore some lost luster to their political standings at home.

The superpower summit is not happening at the best of times for Mikhail Gorbachev. For the first time since he became Soviet leader, his grip on power is under question in some quarters of Soviet society. His country is bracing for several years of economic changes, which are expected to bring price rises, a production slowdown, and perhaps even negative economic growth.

This means he will want to achieve at least two things in Washington. One is very tangible: the signing of an intermediate-range nuclear forces (INF) accord. The other is less tangible, but also highly important: to project himself to the West and his own people as an outward-looking, persuasive, and successful leader.

In the long run, major cuts in nuclear weaponry will yield important savings and allow the Soviet leadership to shift industrial resources from defense to the civilian sector. In the shorter term, a breakthrough in arms control will help prove the radical reformers' contention that risk-taking in policymaking pays off. In the case of arms control policy, Soviet officials say, Mr. Gorbachev took risks with his earlier moratorium on nuclear tests and some of his current concessions on INF. In domestic policy, Gorbachev is taking risks with his push for fast, radical transformation of the Soviet economy. Not all the leadership is happy with risk-taking; many seem concerned that the risks may ultimately prove counterproductive.

The summit will be important for Gorbachev's political standing at home. Public opinion plays a much less important role in the Soviet Union than it does in the West, but it has become a factor in Soviet politics under Gorbachev. Significant segments of Soviet society are beginning to voice doubts that Gorbachev has the full support of more conservative Communist Party leaders, notably Yegor Ligachev, No. 2 on the ruling Politburo.

In a Dec. 1 address, Politburo member Alexander Yakovlev - viewed as the principal ideologist of radical reform, and a member of Gorbachev's summit delegation - defined what he described as ``the most important dividing line'' in the debate over far-reaching reform. On one side, Mr. Yakovlev said, were the supporters of ``really radical restructuring.'' On the other were those who stood for the ``unhurried perfection [and] partial modernization'' of the present system. Yakovlev would clearly place himself and Gorbachev on the side of radical reform. There is, however, a growing perception among Soviet observers that Mr. Ligachev belongs to the second group.

Six months ago, it would have seemed inconceivable that Gorbachev would need to reassure the Soviet people that he was firmly in charge. But some time in the fall, his image began to fade slightly. Discordant speeches by Ligachev and Viktor Chebrikov, the chief of the KGB (secret police), gave rise to speculation that senior party leaders were unhappy at the fast pace of change. The Yeltsin affair, which surfaced in late October, hurt Gorbachev seriously. The affair, in which Boris Yeltsin criticized the slow pace of reform and was subsequently removed as Moscow party chief, has been followed by a sudden surge in informal public comment, most of it negative, about the prominent role played by Gorbachev's wife. A number of Soviets, both officials and non-officials, say they believe the murmurings were in part deliberately inspired by Gorbachev's political opponents.

Gorbachev, like President Reagan, did not win rave reviews with his pre-summit television interview.

Moreover, to some degree he was upstaged by Ligachev, who last Thursday noted in an interview with the French daily Le Monde that he presides over meetings of the Secretariat, which handles top policy issues between Politburo meetings. The comment startled veteran Western observers, who had assumed that Gorbachev himself would chair such an important body. Soviet officials have been at pains to stress that Ligachev's role was neither new nor startling. Nikolai Shishlin, a deputy chief of the Central Committee's propaganda department, said Saturday that Ligachev had been chairing the Secretariat since April 1985.