When it first convenes in January, the reconstituted District of Columbia City Council will be a body in search of new leadership to replace its two most influential members who are departing.

With Sterling Tucker leaving city government after four years as council chairman and Marion Barry moving up to the mayor's office, the council will have just one new face and a new - but familiar - presiding officer.

The new face will be that of Betty Ann Kane, 37, a Democrat who was elected to an at-large seat Tuesday. She is the first white person elected to citywide office since the advent of home rule in 1975.

The new presiding officer will be Arrington Dixon, 35, who is currently the Democratic council member from Ward 4 who was elevated to the council chairmanship in Tuesday's election.

Kane, interviewed yesterday, said she is anxious to be assigned to the council's finance and revenue committee and hopes to be considered as its chairman, replacing Barry. For the leadership post, she can expect stiff competition from a holdover committee member, David A. Clarke (D-Ward 1), among others.

Kane currently serves as finance committee chairman of the D.C. Board of Education.

Dixon, who should have an influential voice in committee assignments and chairmanships, was silent on the matter yesterday.

Preparing to fly to Dallas for a meeting of Southern state legislators (reflecting the District's unique status as a city-state), he would say only that he wants to talk to other council members about their hopes and plans.

Like Congress and many state legislatures, the D.C. Council adjourns at the end of each two-year session, as it will do next month. All pending legislation that has not been enacted will die, subject to reintroduction in January. The council will have power to adopt new rules, revamp its committee structure and choose new committee heads.

The election of Barry as mayor and Dixon as council chairman will open up two important chairmanships - Barry's current post on finance and revenue, and Dixon's as chairman of the government operations committee.

Finance and revenue deals with all legislation on taxation and city revenues. Government operations has wide-ranging responsibility for laws dealing with city administration and personnel.

The council currently has nine standing committees. In addition, the council's full membership, normally 13, sits as a committee to deal with the city budget and the confirmation of people appointed to various city posts by the mayor.

When it reconvenes in January, the council will have two vacancies - the seats now occupied by Barry and Dixon, who must resign to move to their higher offices. Both seats will be filled at a special election May 1. Barry's seat will be filled temporarily, however, by appointment of the D.C. Democratic State Committee.

Kane will occupy the seat now held by Douglas E. Moore, who made an unsuccessful run in the September Democratic primary for the council chairmanship.

Kane, born in New Jersey and educated in New England, is married to a government lawyer. Their two children attend neighborhood schools on Capitol Hill. She is a former teacher and staff member of the Museum of African Art.

In the primary election campaign for the council seat, in which she defeated eight opponents, Kane stressed her work as the school board's finance chairman.

As a new council member, she said yesterday, "My first priorities are getting the budget process and the revenue-producing capacity of the city beefed up."

By that, she explained, she means that the city should press for budget autonomy from the federal government, should be sure that its revenue-raising procedures are tightened up without raising tax rates, and should move as fast as possible toward selling municipal bonds instead of continuing costly borrowing from the U. S. Treasury.

She also said she plans to press for the creation of a study commission to explore ways to put school financing on a solid footing.

Dixon, who as a committee chairman has sought consensus among members rather than pulling the levers of power, is a Washington native who is both a lawyer and a computer specialist. He is a conservative dresser, fond o fgray pin-striped suits.

As chairman, he will have the power to choose several top-ranking council officers, including a successor to council secretary Robert A. Williams, a Tucker appointee.