The tax assessor of Cook County checked into a room at Chicago's Airport Hilton for all of two hours, to do his politicking in private.

Thomas Hynes, the assessor, is the second most powerful elected official in the county organization. But he was thinking, on that Sunday morning in November, that it would do him nothing but harm to be seen in the company of President Carter's national campaign manager -- even for a quick cup of coffee in the coffee shop. "I could explain this to you for two hours and you still wouldn't know how it can hurt us if we go against the mayor," the assessor told Tim Kraft, the Carter campaign manager, when they met upstairs in the hotel room. "You won't understand how rocky this can be and how mean it's going to get."

That understanding would come with itime.

The winning of Thomas Hynes was just one of a number of quiet operations in which the Carter people worked early to see to it that Sen. Edward Kennedy would have no solid late option after early defeats in Iowa and New Hampshire.

On Tuesday, Massachusetts holds its presidential primary. While Kennedy is favored to carry his home state, the victory is expected to do little to convince people elsewhere that he is indeed a threat to the president for the nomination. The following week should be Carter's, with three Southern primaries in Florida, Georgia, and Alabama; Kennedy will make only a minimal effort in these contests. Then comes the Illinois primary on March 18, and New York and Connecticut on March 25.

The Kennedy campaign has announced that it now will go for broke -- literally -- in the hopes of salvaging the senator's candidacy by winning the Illinois or New York. But Carter's lieutenants have succeeded in stripping away much of the support in those two states that might have been expected to go to Kennedy.

In the meeting with Carter's campaign manager, Hynes was wrestling with the idea of going with Carter and breaking with Chicago Mayor Jane Byrne, who was supporting Kennedy. He kept asking Kraft: "Are you guys definitely in this race to stay?No matter what happens?"

Kraft made a telephone call to Camp David. Hamilton Jordan telephoned Hynes with assurances that afternoon. The president telephoned him that night.

Eventually, the county assessor became a Carter man. But as one of the president's strategists observed, "Tom Hynes paid dearly."

When Hynes stood up at a meeting of the Cook County organization and suggested that it would be premature to endorse Kennedy or anyone else at that time, a protege of his -- the deputy director of the mayor's office of senior citizens -- was informed within half an hour that his services no longer were needed.

Hynes formally endorsed Carter later in November, and after that about a dozen of his friends and associates on the city payroll learned from their bosses that they, too, had become expendable.

"Tom Hynes represented a genuine and powerful crack in Jane Byrne's organization," says Kraft. "It made supporting Carter respectable for many in the organization."

The cracks are there. Of Cook County's 80 Democratic committee members, only 16 actually are running as Kennedy delegates. A number of prominent names have chosen to sit out the contest.

This is crucial because in Illinois delegates are elected on their own and it is considered ilmportant to have well-known, vote-getting names on presidential delegate slates. Some of the big names who have chosen not to run this time are Rep. Daniel Rostenkowski, State Sen. Richard Daley (son of the late mayor of Chicago), and Ed Kelly, head of the Chicago park district.

"Kennedy is paying now because he was not organized early in Illinois," says Robert Torricelli, who did the early organizing for Carter in the state. "If Kennedy had his organization in place early, perhaps he really could have made his big stand in Illinois."

Organization alone, however, probably could not do the job for Kennedy in Illinois. In recent weeks, this state apparently has swung strongly toward Carter. The latest Chicago Sun Times poll gives the president a whopping 5-to-1 lead over Kennedy.

Carter's support is very strong downstate, but he also has significant and solid support in Chicago -- including backers within the city's ethnic Catholic neighborhoods, where voters have had a strong affinity for the Kennedy family but are today more conservative than is Edward Kennedy.

Many of the early courtship operations in Illinois were conducted for Carter by Vice President Mondale and his aides, such as James Johnson and Torricelli, who left the vice president's staff early last year to work in Illinois.

While Carter officials are starting to feel comfortable about Illinois they remain concerned about New York -- despite what has been a stampede of prominent Democrats who have been racing to endorse the president while it is still early enough for Carter to be appreciative.

The concerns of Carter officials spring from a variety of factors.

There is, for example, the feeling of Tim Kraft: "What I know about New York and its political dynamics, you could put on the head of a pin. Neither side is going to get blown away in New York, in my opinion. Kennedy might even win there, but it's not going to be large enough so that people will say that he got well in New York."

There is another reason for the Carter concerns about New York that is not often spoken publicly but is real, just the same. It is that New York Democrats have never really felt comfortable with Carter, and vice versa.

In 1976, just after he defeated all other candidates in the Illinois primary, Carter finished fourth in New York. Now Carter's polls are said to show him holding a moderate lead in the state, but his advisers are concerned once again that his potential vote there appears "soft."

This uneasiness could be detected as late as last October in the undercurrents at a reception attended by the president and most of the prominent Democrats in New York State. He was their president, and they clapped for him and smiled at him and shook his hand.

But when a reporter walked among them, they would say privately how they were undecided, and ever looking toward a Kennedy candidacy with enthusiasm.

"To look at his tonight, you would think Jimmy Carter was the greatest thing since 7-up," Rep. Joseph Addabbo (D-N.Y.) said at the reception. "But if Ted Kennedy was next door tonight, it would be a different story in here."'

Among those who were undecided and leaning toward Kennedy on the night of the reception was the Nassau County Democratic chairman, Stanley Harwood. But in December, Harwood endorsed Carter.

Harwood had been among those public officials who had been contacted repeatedly by Carter operatives, ranging from Joel McCleary (who left the White House staff to organize the state for Carter and has been given high marks for his efforts by a number of the politicians in the state) to the president himself.

"What happened to me is what happens to a lot of people here," Harwood said. "I had always had strong feelings for the Kennedys and expected that Ten Kennedy would logically be president someday. But I developed this growilng feeling that he just did not have it -- and that he was no electable because of his personal problems."

Most of New York's prominent elected officials have endorsed Carter, including New York City Mayor Edward Koch, former mayor Abraham Beame, Lt. Gov. Mario Cuomo, the mayors of Albany, Syracuse, and Buffalo, and 50 of the 62 county chairpersons of the Democratic Party.

Two significant endorsements have been withheld. Gov. Hugh Carey and Sen. Daniel Patrick Moynihan have held themselves neutral, and Carey has in the process won the anger of a number of presidential advisors who consider him ungrateful.

The New York state Democratic chairman, Dominic Barannello, had been withholding his endorsement, waiting to see which way Carey moved. Finally, when it became apparent that Carey would not move, Barannello came out and endorsed Carter on his own, two weeks ago.

Barannello said that he acted after canvassing the state and finding "widespread support" for Carter. He said he also has been the recipient of constatn urgings from the president and his advisors.

"No one said I was going to lose my citizenship," Barannello said of the pressure that was exerted. "Actually, I sort of enjoyed that period of seduction."

The Kennedy campaign in New York, meanwhile, has done much of its political fighting internally. Several of the small collections of name Democrats who once signed on to help run the campaign have left.

Joseph Crangle, a top national Kennedy campaign offical who is also the Erie County, N.Y., Democratic leader, concedes that in the rush for public officials, Carter has out-organized Kennedy in Illinois and New York.

"The most conservative human being is your political leader," said Crangle. "His past is loyalty-to-the-president and he's got his blinders on. cBut in New York, when you have the names Carter and Kennedy on the ballot, you don't need a guy in a club with a palm card to tell you how to vote. You can say we don't have the organization for targeting and turnout . . . .

"But turnout is up everywhere.Our people will come out. We just have to bypass the organizations and go directly to the people."