The Prince George's County school board, declaring extreme reluctance, approved a series of drastic budget cuts last night that board members said will result in layoffs of up to 900 employes, about 450 of them teachers.

The cuts followed 10 hours of agonizing public deliberation over two days. Board Chairman Doris Eugene was unable to read the budget resolution before the final vote, struggling unsuccessfully to hold back tears.

"We have worked so hard and come so far in serving the needs of our special kids," she said after the meeting, "but no one else seems to care."

School Superintendent Edward J. Feeney, who produced the list of cuts that was adopted, said: "We are undoing tonight some of the things that we spent years doing. It is tearing me apart."

The cuts, totaling $31 million, were forced by limitations on the property tax contained in the county charter. Among other things they will force more elementary school pupils next fall to walk to school, where they will find larger classes, old books and fewer librarians, music teachers and custodians.

Among the personnel cuts agreed to last night were:

* 168 secondary teachers, increasing class size by two students per class, and saving $3 million.

* 92 elementary teachers, increasing class size by two students (to an average of more than 30 students a class) saving $1.7 million.

* 55 elementary school librarians, leaving one for every two schools, saving $1 million.

* 74 elementary music instructors, eliminating instrumental music in elementary schools.

* 27 secondary vice-principals and administrative assistants, saving $494,000.

* 163 bus drivers, custodians, and maintenance workers, saving $2.3 million.

* 55 secondary library aides, saving $619,000.

Additional cuts include ending school bus service for elementary and secondary school students who live within 1 1/2 miles from school, eliminating $2 million in planned maintenance and improvement of school buildings, and eliminating any purchase of new classroom text books and library books.

Spending on school athletic programs will be cut 40 percent, saving $400,000.

Another savings of $1 million will come from "expenditure control," which assistant superintendent Edward Felegy described as "a hope and a prayer" that the school system "will squeeze out another $1 million next year somehow".

In March, Feeney had asked for $337.4 million to run the school system -- 13th largest in the nation. As in the past three years, the budget was condemned by County Executive Lawrence J. Hogan, who reduced it to $300.9 million. Hogan said he placed a greater priority on public safety employes than public education, but insisted that the reduction could be accomplished without massive layoffs, particularly if the school board renegotiated what he termed an excessive contract agreement with the teachers.

As in past years, school officials hoped that the all-Democratic County Council would make up a large part of the cut made by Hogan, a Republican -- possibly reducing increases Hogan made in the fire and police budgets.

But the council, restricted by the TRIM county charter amendment that limits property tax levels, restored only $5.4 million, leaving the school board members to decide what people and programs to cut.

At one point, at the urging of board member Norman Saunders with the backing of Leslie Kreimer, the board seriously considered eliminating interscholastic sports and school clubs. to save an additional $1.4 million for use in other areas.

Members Al Golato and Angelo Castelli reminded their colleages that the sports program keeps many would-be dropouts in school and out of trouble.

"I shudder to think about what will happen to Prince George's County after 3:30 every day with everybody on the street," said Castelli.

"They've already hired more police to take care of it," said Kreimer.

Despite Hogan's March warning that the school budget request was unfundable, the school staff waited until four weeks ago to draw up the list of budget cuts adopted last night. School board members did not see the entire list until Tuesday.

"We were hoping for something. . . anything," school spokesman Brian J. Porter said yesterday. "The money is there, but they gave it to other agencies, and they told us to take a walk."

In making repeated apologies for the cuts, board members held out a slim hope that school employe unions might make some concessions before the June 17 deadline for mailing layoff notices. Such concessions, they said, might save jobs. The county school system now has 7,309 teachers and 3,063 other workers, exclusive of administrators.

The teachers' union executive council will meet this afternoon, but union president John C. Sisson bitterly denounced the board's expectation and criticized its inability to rearrange the budget to save more teachers.

Concessions "will be discussed," he said. "Had the board cut the area offices, had the board cut extracurriculaar activities, I think that would have been a show of good faith that would have left our members more inclined to consider alternatives." Sisson has maintained that the contract, which provides for 6 percent raises this year, cannot be touched.