Montgomery County, one of the two richest counties in the nation, is about $700,000 behind in payments to scores of private schools that educate mentally and emotionally handicapped students the county school system has no place for.

Because of the delays in the county's payments over the last few months, some of these small, non profit schools, in order to meet their payrolls, have been forced to borrow or have dunned students' parents, asking them to make up the difference until the county can pay.

The situation, however, is nothing new to the schools. Montgomery County is merely the latest local jurisdiction to have left the private schools from New England to Florida hanging, sometimes for months, before their bills are finally paid.

"We experience this from all the counties," explained Pam Wallace, of Alexandria's School for Contemporary Education.

"We had a problem earlier this year with the District," she said. "We had a problem with the Prince George's County schools. We didn't even have as much of a problem with montgomery as we did with some of the others."

Right now, however, their problem is Montgomery County's bill of about $22,000. At the Leary School in Falls Church, Montgomery County has had an outstanding bill of about $40,000 since February. The county has promised these schools and scores of others that they will be paid on about June 30.

Everyone involved - private school administrators, local officials, state officials and parents - agree that late governmental payments are an incessant problem. When it comes time to explain the cause of the problem, however, the different groups begin a merry-go-round of finger-pointing, each passing along some of the blame to the next.

The schools blame the local governments for not appropriating enough money to pay their bills; the local governments respond that hefty and unexpected tuition increases over the last year had rendered their planned budgets inadequate.

Maryland's State Department of Education has questioned the need for some of the tuition increases, which have ranged from 6 per cent to 100 per cent.

The parents tend to feel the government is at fault. "I blame the state. They're tardy. They're dragging their heels. Some of the waste in the state should be turned over to useful purposes like this." said Raymond Ramsay of Wheaton, whose son Philip, 16, is a student at the Leary School.

Maryland officials concede that state actions this year contributed to the problems Montgomery County has had this spring - and Prince George's County had last fall - in meeting its obligations.

Under state law, individual counties must pay for private education for their students at triple the rate it would drop from the normal $97 to costs to educate an average public school student. Under this formula, Montgomery was obligated to pay $4,920 for each private placement student this year; for Prince George's the figure was $3,891.

Private placement costs beyond this, without limit, are paid by the state - and tuition can range as high as $20,000 and more for some students in residential care.

Last year, Maryland budgeted about $2.4 million for the private placement of about 2,000 students, 950 of them from Montgomery alone. That budget represented a 75 per cent increase. But at the same time private tuition went up an average of 14 per cent, according to state officials.

When stae officials realized last summer how under-budgeted they were, they directed local school systems to repeat the annual ritual of form-filling, which in many cases had been completed, with more stringent forms to be used.

One state official defended the need for the re-evaluation, saying that some students had been shifted into unnecessarily expensive programs, and in other cases schools were billing the state for students' riding lessons, laundry costs and pocket money.

Problems with late payment have recurred with alarming regularity over the past three years, since a series of court decisions and laws have required that Maryland, Virginia and the District pay for the appropriate private education of those students who can't be taught in the public schools, officials said.

Many of the schools enrolling these students are small, with student bodies hand. "We had to borrow $17,000 last month to meet payroll," said Harold Clarke, business manager at the Leary School, which is currently owed about $40,000 by Montogmery.

"We predicate cash flow coming in at specific times. We don't build in a great margin of error . . . We don't have a lot of money to fall back on," Clarke added.

For the Schools for Contemporary Education, whose tuition increased 6 per cent this year, "the situation will become critical in July if something doesn't happen with Montgomery," said administrative director Sally Sibley.

Yet, for several months lst fall, she said, "Prince George's actually withheld all the money, their share and the state's, until they knew what the state was going to do."

At that same time, "Montgomery went ahead and paid their full share - but now they say they don't know when they'll have the money to pay the rest," she said.

Dr. Enzo Monti, supervisor of placement for the country's schools, agreed with state officials that "it's a basic matter of not providing the money in the budget for what the costs would be." The county planned on spending $4.2 million for private placement, but the bills have been closer to $5 million.

"It's been a matter of robbing Peter to pay Paul," Monti added. "I'm very sympathetic to the schools - they're in a tough position . . . The villain is the same one you hear of everywhere - nobody has enough money."

The Leary School's answer to the problem was to send out letters to the parents of Montgomery County students last month 'requiring" them to chip in until Montgomery had paid its bills.

For Raymond Ramsay, this short-notice bill for nearly $1,700 was "flabbergasting." His wife Virginia added, "You just don't pull out $2,000 like that . . . but they seemed to sound quite desperate in that letter."

Later, the county informed the Ramsays and other parents that they had no obligation to pay, and the county would find a way to get its bills paid by July.

Next week, the county Board of Education is asking the County Council for a special supplemental appropriation of about $600,000, in part to make up its delayed payments to schools like Leary.