Mayor Harold Washington and his powerful adversary, Alderman Edward R. Vrdolyak, met briefly today but failed to repair the schism that threatens to stalemate the city government after the mayor's startling defeat Monday by a majority of white aldermen led by Vrdolyak.

Neither had anything to say about the 15-minute meeting in the mayor's office at noon.

Vrdolyak, whose nickname is Fast Eddie, outsprinted a horde of City Hall reporters three flights up to the mayor's office from his alderman's suite on the second floor, and then again on the way back down.

"Have a great day!" Vrdolyak shouted to the reporters bombarding him with questions. He had one himself Monday.

In one of the strongest shows of political muscle since the Democratic Party took control of the Windy City's government 50 years ago, Vrdolyak produced a 29-vote majority on the 50-member City Council, installed his supporters as key committee chairmen, turning out the new mayor's choices, and altered council rules to make it much tougher for Washington to undo what was done.

This show of strength left Washington, whose election as the city's first black mayor in a racially divisive campaign drew national attention, struggling to find a compromise with the white majority.

He maintains that the Monday council meeting was adjourned legally when he banged his gavel only five seconds after it had convened and stalked out of the chambers with his 21 aldermen, and that Vrdolyak's subsequent actions are void.

"Mr. Vrdolyak and some few of his supporters don't understand that there will not be business as usual, that there will be reforms," Washington said.

But a principal Vrdolyak lieutenant, Alderman Edward Burke, showed no interest in compromising.

"It will take a two-thirds vote to rescind anything that was done yesterday," Burke said today.

He was voted the new chairman of the powerful Finance Committee, replacing a Washington ally, Alderman Wilson Frost, a black whose ouster from the committee chairmanship is one of Vrdolyak's major conditions for an agreement, according to City Hall sources.

Political analyst Milton L. Rakove said the Vrdolyak aldermen "just voted the way their constituencies wanted them to."

Of the 29, four are from wards won by Washington, but only one, Alderman Miguel Santiago, a Hispanic, was considered a possible Washington ally in a showdown.

Unless Washington and Vrdolyak can find a way to compose their differences, Chicago may be in for four years of government by stalemate.

The first milestone is Friday, the date Vrdolyak's majority set for the next council meeting. But Washington says he and his aldermen will not attend.

Meanwhile, Vrdolyak has enough votes in the council to make law, but Washington can veto much of it. Some of his aldermen said today that he would not appropriate the money to run the 29 committees set up by Vrdolyak.

While Washington says he is ready to compromise, City Hall regulars question his bargaining power as an anti-patronage mayor dealing with a machine that sees municipal government in terms of patronage jobs for friends, relatives and party regulars.

Timothy C. Evans, one of 16 black aldermen who are the core of Washington's strength, said that "There is room for compromise and the parties realize this. There is room to negotiate."

Later, he added, "in the broadest sense, the city must survive. There are major financial problems, transit problems, the schools to be handled."

In the end, this may be the only direct appeal Washington can make to Vrdolyak.

With his self-styled reform administration less than a week old, the pressure is on Washington to find a way back from the crisis.

On Monday he threatened to take the issue to court.

Today, however, the local media reported that parliamentary experts are doubtful that his position would hold up legally.