Moments after the Alexandria City Council recently decided who would fill three seats on that city's nine-member School Board, Linda Burch's cries of indignation could be heard as she marched out of the council chambers with her flustered husband in tow.

Burch, a well-spoken mother of two sons in Alexandria schools and an active member of the PTA, did not receive a single vote from the council in her second attempt in two years to become a member of the board.

"It's political isn't it?" she asked just before stepping into an elevator. "That's why I was out of the race before it was over."

That night Burch's question went unanswered. But last week, following the installation of incumbents Lou B. Cook, Mary Jane Nugent and first-timer Rhonda D. Hill, Alexandria politicians and others on or close to the School Board, said, quite matter-of-factly, that the answer to Burch's question is yes.

Although Virginia does not permit any locality to have an elected school board, the forces in Alexandria that determine who actually gets the three-year, $125-a-month board member jobs, are nonetheless political.

"It's still a very, very political position," said the board's new chairman, Judith A. Feaver. "Taking the election out of a school board doesn't mean removing a school board from politics."

For example, this year the council was pressured, many of its members say, to appoint a third black to the board.

More than 48 percent of the school system's 10,500 students are black. And, says Ferdinand T. Day, a former black chairman of the board in the early 1970s, "with the growing percent of blacks in the school system, there should be a corresponding growth (of the number of members) on the School Board."

When the black community, which is heavily Democratic, demands from a heavily Democratic council in an election year to see an additional black face on the School Board, political considerations had to be made. (Democrats hold five of the seven votes on the council.)

Accordingly, the field of 12 candidates, which included two blacks, began to narrow quickly. As Republican council member Carlyle C. Ring Jr. put it, 10 of the 12 candidates, who were white, were no longer "qualified" because one of the seats this year had to go to a black.

Ring says it was a matter of "balance."

"For harmony and community support, we need balance. We ought not say that we want a black seat or a Jewish seat or an Alexandria Taxpayers Alliance seat," Ring says. "Nonetheless, we recognize that to marshal the support and arouse the enthusiasm from all the elements of the community, you need to have all the elements of the community represented on the School Board."

Most of the council, including Mayor Charles E. Beatley Jr., agrees. But when it came to deciding which of the two blacks--A. Melvin Miller or Rhonda D. Hill--would be selected, the issue turned from race to partisan politics.

In Virginia, where voters don't declare their party affiliation when they register, it's often hard to tell the Republicans from the Democrats. In the case of Miller, an official of the Department of Housing and Urban Development who says he is a Democrat but is perceived by many on council to support Republican candidates, the confusion has made him widely unpopular with Democratic council members.

In each of his last three attempts to get on the School Board, Miller has been stopped by Democratic votes. The council was looking for a black they could live with politically.

Feaver says, "They (some council members) had already promised that when a strong black candidate for School Board came forward that they didn't have political problems with, like they did with Melvin Miller, that person would get their support."

Hill, 29, a Fairfax County school teacher, was encouraged by people close to the board to run. Soft-spoken, with a masters degree in special education, Hill was considered well-qualified and, more importantly, politically acceptable. She keeps her politics to herself.

After a tie vote between Miller and Hill, the council on a second ballot voted 4 to 3 for Hill. Miller's votes came from the council's two Republicans, Ring and Margaret B. Inman, and the only black council member, Lionel Hope, who said Miller was the most qualified.

Incumbents Lou Cook, who was the outgoing board chairman, and Mary Jane Nugent were reappointed.

Feaver and others close to the board said Cook largely was selected on her two-term record. And Nugent, who Feaver says is a good and thoughtful board member, had the right credentials. She is the legislative aide to Democratic State Del. Bernard Cohen.

Some council members acknowledged Burch was a good candidate, but so were many of the others. And political debts and loyalties to the other candidates sapped much of Burch's potential support. For instance, candidate Jean Swersky, who is deputy chairman of the city's Democratic Committee, reminded Democratic council members she had worked hard to put them in office.

Still, two of the Democrats she's credited with helping the most, Donald C. Casey and Lionel Hope, were the only council members to vote for Swersky.

As for incumbent J. Harvey Harrison Jr., a conservative Republican, "he was the odd man out," Feaver says. "The majority of the board felt the political pressure to make a black appointment rather than to maintain a person from a political element that is no help to them in an election."