AS JIMMY CARTER'S bandwagon hits bumpy terrain, the 86 delegates Chicago's Mayor Daley holds in the name of Illinois Sen. Adlai E. Stevenson '52 begin to look increasingly significant. The vision of taking all those delegates in one fell swoop by choosing Stevenson as his running mate, must loom very large in Carter's mind as he creeps towards the 1505 votes needed to nominate.

Despite their attractiveness, Sens. Walter Mondale and Birch Bayh just don't hold strong hands in the delegate numbers game--the latter spent $1 million for one Massachusetts delegate. Another often-mentioned vice-presidential possibility, Sen. John Glenn, of Ohio and Outer Space, is a political neophyte--something the Carter campaign abounds in--and unless the delegate situation is really desperate, constant primary bickering between Carter and Mo Udall would preclude the choice of the Arizona Congressman. And a Carter-Jerry Brown would be unbalanced in ideology and constituency which would mean sitting this one out for many Democrats.

This scenario, which assumes Jimmy will come out on top, leaves Stevenson as the only one remaining on the pundits' list of vice-presidential possibilities. He provides the old venerated name for Carter's new flashy one, the establishment ties for Carter's anti-Washington constituency, the urban Midwest appeal for Carter's rural Southern one. Most importantly, Adlai finds himself in the unique position of simultaneously representing the party bosses like Daley, plus the liberals who admire the Senator's progressive legislative record and still tingle when they remember his father's greatness. This year the Daley wing is the more important of the two and in that sense Stevenson would represent Carter's insurance against another McGovern debacle. After being ejected from the 1972 convention in a credentials dispute, a triumphant return by Daley would dramatize the reunification of the party's divergent elements--something Carter obviously needs.

Looking ahead to the fall, something else Carter needs is to carry Illinois. Every election since 1916 has found that state on the winning side, and Daley's superhuman efforts to put Illinois in Kennedy's column proved decisive in 1960 (to the chagrin of Republicans suspicious of "ghost voting" illegalities). Ford is very strong in Illinois--besides Michigan it was his most significant win--so if Carter expects to come away with those 25 electoral votes, he'll need the Mayor's whole-hearted support.

But Stevenson could help the national ticket in more general ways. Because he realizes he lacks his father's eloquence and wit, he concentrates instead on making his speeches rhetoric-free and responsive to issues--a trait in short supply this year. Stevenson, who some say knows the issues better than his father did, is given high marks for his performance in the Senate. His well-recognized participation in the drafting of the aid package for New York City would help Carter in stricken urban areas and a knowledgability in foreign affairs seems to run in the family. On energy, Stevenson is considered one of the best informed members of Congress. These are attributes that could bring thoughtful Democrats to swallow a Carter-Stevenson ticket.

SO CHOOSING Stevenson makes some sense. Whether in this strange election Carter will pay attention to what makes sense is a different question. What Jimmy must pay heed to, however, is clout--and that's a word Richard J. Daley put in the American vocabulary. If Daley so desires, he can bring a tremendous amount of pressure to bear on Carter to pick Stevenson. "Hizzoner da mare," as he is known to his Chicago friends, possesses hundreds of I.O.U.'s just waiting to be collected on. For years, Democratic politicians from across the country have come hat in hand to Daley's office and one call from the kingmaker this summer could send any number of them scurrying over to talk to Jimmy about Adlai. Ted Kennedy, for one, is willing to do just about anything Daley asks of him. Carter understands all of this and although he is smart enough not to make any committments yet, he has been courting Daley and Stevenson for some time. In 1974, when the Mayor had a stroke, Carter invited him to Warm Springs, Ga. to recuperate.

Thus, in a peculiar way, the question really becomes one of what Daley wants to do, and this goes to the heart of the Daley-Stevenson relationship. Daley likes national prestige and the extra funds for Chicago that would result from having a hometown boy in the number two spot, but in truth he probably doesn't really care very much whether or not Adlai Stevenson the man, is vice-president.

The association between the two has not been a close one. Daley came up the hard way--to this day he lives in the old Irish-Catholic neighborhood in back of the stock yards on Chicago's South Side. It takes a lot of muscle to run a city like Chicago for over twenty years and that strength comes from a well-oiled patronage machine run by tough-talking, bowler-hatted ward bosses--Daley's own kind.

Adlai Stevenson III didn't come up through the precincts. His great-grandfather was vice-president under Grover Cleveland and in 1952, when his father first ran against Eisenhower for president, young Adlai was between his undergraduate and law school days at Harvard. During those years, Daley never cared much for the late Governor Stevenson and many of the Mayor's lackeys adhered to the current "Adlai the Egghead" sentiment. After losing twice to Eisenhower, the Mayor thought Adlai II had had enough and in 1960 he dumped him in favor of John Kennedy--quite an embarrassment to say the least.

Daley and his power base--the Cook County Democratic Central Committee--usually determine who will represent the party statewide and in 1968 when Adlai III went before the slatemakers asking to begiven the nod to run for the Senate, Daley said no go--Stevenson was against the war. After the bloody convention that year, Stevenson, then state treasurer, criticized the Chicago police and called the Daley machine "feudal."

But in 1970, after Everett Dirksen's death opened a Senate seat, the two made peace and their relationship is now polite, above-board and very business-like. Stevenson needs Daley to get re-slated by the Democrats for the Senate and carry Chicago; Daley needs Stevenson to lead the state ticket and sweep in other Democratic candidates on his coattails as he did in 1970 and 1974.

AND THAT IS exactly why Richard Daley needs Adlai Stevenson this year. Michael Howlett, secretary of state and a Daley man, knocked off anti-Daley incumbent Gov. Dan Walker in the March primary, but this fall Howlett faces a very tough race against the Republican candidate: former U.S. Attorney James R. Thompson. Thompson is the reformer who has sent dozens of Daley's closest friends and associates to jail in recent years and the mere mention of his name is enough to make the Mayor and his henchmen apoplectic.

Daley is a very personal Godfather-like politician and when something touches him closely he does not easily forget. County office-holders, police captains, aldermen, the Mayor's own press secretary and even a former Illinois governor, are among Thompson's victims. Daley's floor leader in the city council just went behind federal bars last month. All of this combined with the fact that he wants a friendly face in the statehouse after battling Walker every inch of the way for four years, could make the Mayor go to any lengths to destroy Thompson in November. Howlett defeated Walker by crushing him in Chicago, but downstate Walker carried almost every county. Howlett will need help down there if he is to triumph over Thompson. Stevenson is very popular in southern Illinois. If Daley honestly thinks his name on the ballot will give Howlett a boost, he might do just about anything in his power to get Carter to choose Stevenson.

Adlai himself would have some input, of course; even now he is no Daley puppet. Stevenson is known to lean toward the Humphrey-Muskie wing of the party (Daley, on the other hand, dislikes Humphrey). Additionally, there is some reason to believe Stevenson might not yearn to be vice-president. But providing he would accept, it is conceivable Daley might try to pull the whole thing off.

Then again, no can ever tell what's going through the mind of that little rotund man on the fifth floor of Chicago's city hall.