The dollar rebounded yesterday from its steep slide of recent weeks, thanks to unusually strong purchases of the American currency by the Federal Reserve and foreign central banks, and the stronger dollar helped send U.S. stock prices sharply upward in the first trading day of the New Year.

The Dow Jones industrial average scored a 76.42 point gain to close at 2015.25, a 3.94 percent rise on the day. Both in terms of points and percentage, it was the fourth-best advance in stock market history. The other three came last October after the market's Black Monday plunge on Oct. 19.

The dollar's recovery from its low points encouraged investors who had seen it fall steadily in the last two weeks of 1987 to new lows not only against the yen, but also against major European currencies.

The day began with a bleak report that the dollar had fallen to 120.25 yen in Sydney, and after opening at 120.45 yen in Tokyo had set its 12th closing low in 18 days at 121.65 yen.

But reports that the Federal Reserve Bank of New York had joined with other central banks to stop the decline turned things around.

In Tokyo today, the dollar opened at 123.00 yen and continued to gain in early trading, finishing the morning session at 124.07 yen.

In reaction to yesterday's rebound, the Dow Jones industrials gained 55 points in the first hour of trading, and stayed strong the rest of the day.

Other factors also appeared to have helped the stock market yesterday. {Details, Page C1.} But the dollar was the main prop.

"The overt explanation is the dollar," said Larry Wachtel of Prudential-Bache Securities. "There was fear that it would cascade downward."

There was general agreement among dealers that the level of intervention by central banks yesterday had accelerated, but that it had not been enormous.

The Bank of Japan appeared to have made the heaviest commitments, on the order of $1 billion to $2 billion.

Many market experts cautioned, however, that the dollar could easily reverse course and move to new low points if the official intervention diminishes, or if the upcoming Jan. 15 monthly trade report shows no significant decline in the American trade deficit.

Conversely, a favorable report probably will be taken as a good sign by the stock market, traders said.

Indicative of the continued uncertainty -- and of the potential volatility -- in exchange markets was the dollar's reaction in midafternoon yesterday, when Market News Service reported a high-ranking but unidentified Fed official as saying, "We're getting close to the low point of the dollar ... {but} I would expect {it} to go a little lower."

When the Market News Service report became known, the dollar, which had risen from around 1.57 marks last Thursday to a high of 1.5890 marks, fell to 1.5850 marks.

The yen fell from a high of 123.22 yen to 123 yen. In New York at the end of the day, the dollar was at 1.5857 marks and 122.65 yen.

Exchange markets seemed convinced yesterday that they were witnessing a coordinated intervention by the central banks of a somewhat different, and more determined, character than had prevailed in the closing weeks of 1987 -- one that might actually stop the dollar decline.

Market observers had expressed the fear that continued weakness in the dollar, carrying with it a growing threat of new inflation, would force the Federal Reserve Board to boost interest rates, thereby making investing in dollar-denominated assets and securities more attractive.

But the negative aspect of an interest rate rise, economists say, is that it could cause a recession by making it more expensive for individuals to buy homes or for businesses to borrow money for new ventures.

In a day of seesawing dollar movements, French Economics Minister Edouard Balladur was reported to have said the recent agreement of the Group of Seven major nations contained "a secret clause" pledging them to cooperate in the exchange markets to promote stability of the dollar.

Balladur's comment was simply a reiteration of a paragraph of the G-7 communique on Dec. 22 that -- as official briefers indicated at the time -- strongly implied that the central banks would coordinate their intervention policies.

But the markets took Balladur's words as confirmation that yesterday's intervention was perhaps a bit stronger, or reflective of a belief of government authorities that the dollar was at or close to the bottom.

The dollar's rise from its low points yesterday still leaves it below the 126 yen and 1.63 German mark levels that prevailed on Dec. 22 when the G-7 statement was issued.

In that statement, the G-7 -- the finance ministers and central bankers of the United States, Japan, West Germany, France, Britain, Canada and Italy -- said that "a further decline of the dollar ... could be counterproductive by damaging growth prospects in the world economy."

Apart from the implied concerted intervention, cited by Balladur, the pact did not spell out specific actions.

The U.S. Treasury refused all comments on Balladur's reference to a secret clause.

A spokesman, though, called attention to the language of the communique, which said that in the effort to stabilize currencies, the ministers "agreed to cooperate closely on exchange rates."

Shigeru Tokunaga, a foreign exchange dealer for the Fuji Bank in New York, said that one thing helping the dollar yesterday was the report that for the first time during the recent cycle of events, the Fed had intervened in Asian markets, either in Hong Kong or Singapore.

Tokunaga said that direct Fed intervention in Asia would be taken as a sign "that it is the desire of the United States, not just of the G-7" to halt the dollar decline.

Fed officials in New York had no comment on that rumor, but recalled that U.S. central bank intervention to support the dollar with purchases in Asian markets on two occasions last June had been publicly acknowledged.