When Mayor Jane Byrne advised Cook County Democratic Chairman George W. Dunne 48 hours in advance that she would endorse Teddy Kennedy for president, he replied that "anyboby certainly woud be better" than Jimmy Carter -- reflecting the mess of Carter-style politics in vital Illinois.

Dunne's curt dismissal of President Carter suggests the futility of White House operations during the past month. While frantically courting the mayor with lavish attention and federal largess, Carter's long neglect of Dunne and his cohorts doomed him. Even while Byrne briefly flirted with the Carter candidacy, Dunne was moving toward slating the organization's delegates for Kennedy.

That means Carter begins the battle for Illinois' 179 delegates with Chicago in Sen. Edward Kennedy's pocket. Since the president can at best hope for an even split of downstate delegates, his prospects are bleak next March 18 in the very primary labeled essential by his own agents.

Talk from the White House about the necessity of winning Illinois was tied to the slender thread of Jane Byrne's support (which she expressed in a near-endorsement of Carter during his Oct. 15 visit here). Carter's strategy was to contest Kennedy for downstate delegates while counting on the mayor to drive uncommitted Cook County organization delegates into the president's pocket. It was a strategy hopelessly flawed by three misperceptions.

The first was that Kennedy would stand idly by without running delegates (as past candidates, including Jimmy Carter in 1976, have done) while the Cook County organization elected an uncommitted slate. In fact, Kennedy strategists in Washington do not intend in any state to acquiesce in uncommitted delegates, who are susceptible to the lures of an incumbent president.

What's more, in this day of the declining organization, any primary delegates bearing the Kennedy label would be favored here to defeat the uncommitteds. Thus, the organization was faced with the same choice posed more explicitly to Gov. Michael DiSalle of Ohio by John F. Kennedy in 1960: run delegates committed to me, or I'll beat you with my own delegates.

The second misperception exaggerated Byrne's control. Unlike the fabled Richard J. Daley, she does not wear the double crown of mayor and county chairman. While city hall patronage gives her major party leverage, she collaborates with Dunne (as president of the Cook County Board, he is her neighbor on the fifth floor of the city-county building). Some time ago, Dunne privately decided the organization eventually would have to play the Kennedy card.

That was only partly because he realistically recognized the prospect of Kennedy-pledged delegates winning. Dunne, calm and soft-spoken, has smarted under what he has considered three years of neglect by Carter. On the day before his climactic meeting with the mayor, Dunne told us: "We have to look at what consideration we have received from the president. The answer is none."

The third misperception was the belief that pork barrel was sufficient to woo and win Jane Byrne, who entered politics as a Jack Kennedy volunteer in 1960. Practical considerations posed by Dunne only confirmed what Chicago insiders always thought would be her ultimate choice.

Nobody really knows why she went so far in her pro-Carter statements Oct. 15. But three days later, she was miffed by statements from Health, Education, and Welfare Secretary Patricia Roberts Harris attacking Chicago in the school desegregation impasse. The president had failed to silence Mrs. Harris as promised.

Soon afterward, Kennedy brother-in-law Stephen Smith was off to Chicago on one of his innumerable family political missions: to assure the mayor that draft-Kennedy leaders objectionable to her would be downgraded for the campaign. Meanwhile, Teddy Kennedy himself was courting her by phone from Washington. A further prod: state Sen. Richie Daley, son of the late mayor, who is viewed by Byrne as a future challenger, was about to endore Kennedy. Why not beat him to it?

The effect of all this was evident Oct. 24 when Byrne was in Washington unwittingly attending the president's campaign kickoff. She told one fellow Illinois Democrat that she was furious over having come to the capital under false pretenses, unaware that the Carter dinner (which she ducked) was a campaign rally for loyalists. Back in Chicago late in the afternoon the next day, she had her revealing chat with George Dunne.

Dunne was surprised at her timing, expecting the organization would not tip its hand until just before the January filing deadline. But that was nothing compared with the shock of Carter operatives when they learned their Illinois strategy was pure illusion. Carter's Chicago follies only deepen the mystery of just how this incumbent president can be renominate.