The construction of four public schools in Prince William County cannot begin until someone comes up with $370,000, the Board of County Supervisors and School Board agree. But neither group is racing to pick up the tab.

Prince William voters, hoping to relieve widespread classroom crowding in the rapidly growing Northern Virginia area, agreed this month to accept $44.89 million in bonded debt to build the new schools.

The money from the bond sales won't be available for several months, however, prompting a request from school officials to the county board for an advance so work on the projects can start immediately.

County Executive Robert S. Noe Jr. said yesterday that he told school Superintendent Edward L. Kelly that the county government simply doesn't have the money.

The School Board, Noe said, should find the construction funds from its own budget.

Kelly maintains that the school system will have to raid funds it has allocated for other purposes.

"The bottom line is we don't have the money in our construction fund," Kelly said yesterday. "We have to see where we can get it and least hurt the instructional program."

Both sides said it is unlikely that construction of the high school, middle school and two elementary schools would be delayed by the quibble over funding.

Ordinarily both the county government and the school system would have some money on hand for projects that weren't included in the annual budget. What spare funds the schools had on hand, however, have been spent to remove absestos from several schools, Kelly said.

The county government's "unencumbered" funds, meanwhile, have dwindled to only about $40,000 after supervisors -- in the midst of an election year -- slashed reserve funds to cut the tax rate.