In late August, just a week before school was to open, Principal Richard Gastley had everything he needed to get the year off to a perfect start at Prince William County's Springwood Elementary School -- almost.

Gastley had a new $3.5 million school and 13 teachers ready to teach the 319 pupils who planners had predicted would arrive.

The only problem: "We ended up with 420 students on the first day," Gastley said. "We were 99 over the projections."

Springwood was short four teachers.

The school's problem was hardly unique. The Prince William school system -- with more than 36,800 students, the third largest in Virginia -- has been struggling for years to keep up with the staggering population growth in the once rural but now increasingly suburban county.

This year those efforts fell short, raising the question of how the school system will persuade longtime residents and newcomers alike to be committed to raising the funds officials say are necessary to build more schools.

The system ended up with almost 600 more students than projected, and with nearly 1,000 more than were enrolled last year, according to Charles Wildman, head of the school system's planning office.

Fourteen schools are using trailers or portable classrooms to cope with their overflow, and 19 teachers have been hired since the start of the school year.

"It's not a good way to hire," conceded William Cox, Prince William's associate superintendent for instruction. "Once the school year begins, students and parents do not like to change teachers."

Of even greater concern to Prince William educators, however, is the need for money to build new schools that this year's spurt in numbers seems to foreshadow.

"I don't think there is any doubt that we will have to build a new high school . . . and that we will have to continue to build elementary schools as we have been doing," said Superintendent Richard Johnson.

Historically, Prince William has had difficulty passing bond issues. And some educators fear a renewed bout of the longstanding tension between established Prince William residents and suburban newcomers, who bring with them new demands for superior schools.

Prince William is no stranger to overenrollment problems. During the 1970s, many schools there had split-shift class days and year-round classes to accommodate the increase in students. There have been bitter controversies in the past over decisions to bus students miles away from schools near their homes to schools in less crowded parts of the county. A reprieve came when numbers dipped because of growth trends and the decision by the cities of Manassas and Manassas Park to set up their own school systems.

But this year, school officials were presented with 3,000 reasons that the reprieve is over. Namely, the 3,000 new homes that were built in the county the preceding year. That was nearly 1,200 more homes than expected, according to planning official Wildman, a spurt he said was primarily attributed to the strength of the economy. Wildman now projects that the county will continue to grow at this rate for the next several years.

For the immediate future, the education crunch at the secondary school level is the biggest worry. "All four high schools in the most rapidly growing areas of the county are either at or above capacity," Wildman said.

A new high school is on the school system's capital improvements plan, and officials said it is likely that a bond referendum for a new high school -- usually a $20 million to $30 million item -- will go to the voters in either 1987 or 1988. Over the long range, however, growth now taking place at the elementary school level presents the biggest challnege. "The largest part of our growth has taken place at the kindergarten through third grade level."

Prince William has built 10 schools since 1975, most of them less expensive elementary schools that can be built without bond referendums, according to Kristy Larson, Prince William schools spokeswoman. But even the new elementary schools are feeling the pinch of growth. Lake Ridge Elementary School, for example, already has more students than its official capacity this year, only the second year it has been open.

Assistant Principal Gerald Boling said 740 students showed up for the opening day of school, nearly 80 more than had been projected. Classrooms that can be relocated have been set up outside the school to handle the overflow.

But as elementary school-age children move up the system, year-round schools, busing and other makeshift measures that Prince William has used in the past to forestall building new secondary schools will not work, school officials said.

"We don't look to those as solutions," said Associate Superintendent Richard Chapin. "The solution is to build."

Opinions vary on whether public support for construction will be there. But support for a strong educational system seems to have risen as the county has grown.

"Prince William is no longer a rural community. It's a more urban community, and more and more people are expecting a higher level of education," Wildman said.

Other officials are more skeptical. School board member Regis Lacey said the proportion of households with school-age children has been going down, and a large percentage of the parents with children enrolled in Prince William schools are in the military and are not registered to vote in the county.

If Lacey is right, that points to a continued struggle for school officials to keep even with the current growth and to get ahead of the growth that their charts and projections tell them is coming. Said Wildman: "We are doing our best to stay in front of that power curve."