The case has been closely watched by educators. Almost seven million students nationwide receive special-education services, with 71,000 educated in private schools at public expense, according to the federal Education Department. Usually, districts agree to pay for those services after conceding that they cannot provide suitable ones.

New York City pays for private schools for more than 7,000 severely handicapped children because it agrees that it cannot properly instruct them. But, officials said, requests for tuition payments for special education students by parents who have placed their children in private school on their own have more than doubled in five years, to 3,675 in 2006 from 1,519 in 2002. And the cost of these payments grew to more than $57 million in the last school year.

“The trend has been increasing for several years,” said Michael Best, general counsel for the city’s Education Department.

Leonard Koerner, chief of the New York City Law Department’s appeals division, said: “We are very disappointed in the court’s ruling, because it does not require the parents to place their children initially in the public school system. This detracts from schools’ abilities to work with parents for the best possible educational outcomes for children with disabilities.”

But Mr. Koerner noted that the ruling did not set a precedent that would bind all schools in the country, and he expressed hope that the justices would soon consider the issues again. That seems likely. At least one other circuit court has come to a conclusion opposite from that of the Second Circuit.

Moreover, there is another Second Circuit case, from Hyde Park, N.Y., that is already available for review.

The justices last February decided to hear arguments in the New York City case and not the Hyde Park one, but Justice Kennedy’s decision to recuse himself only 12 days before the New York City case was argued created the possibility of the 4-4 tie and could make the court inclined to take up the issue again this winter. The court gave no reason for Mr. Kennedy’s lack of participation.

Both the New York and Hyde Park cases involve interpretations of a landmark 1975 special education law now known as the Individuals With Disabilities Education Act and amendments to it in 1997. The law requires school systems to provide a “free appropriate” public education to disabled students, with individually tailored programs.

The law permits parents to seek public financing for private schools if they can show that the public schools cannot meet their children’s needs.

Mr. Freston’s fight on behalf of his son began a decade ago, when his son, then 8, was found to have learning disabilities.

The city offered the child a coveted spot in the Lower Laboratory School for Gifted Education, on the Upper East Side, but Mr. Freston wanted a smaller setting and put his child in the private Stephen Gaynor School. He won tuition reimbursements through administrative hearings and an appeals board proceeding.

Then the Board of Education sued in federal court, where a district judge ruled that a family could not receive tuition reimbursement unless a child first attended public school. But the Second Circuit court found for Mr. Freston, sending the case to the Supreme Court.

Mr. Freston, who left Viacom with a separation package worth an estimated $85 million, has said he brought the case on principle and has donated his tuition reimbursement to tutoring for public school children.

The Hyde Park case involves a boy born in 1991 to a crack-addicted mother and suffering from learning disabilities. The child’s adoptive parents placed him in a private school after turning down the public school programs designed for him. A federal district judge ruled that the parents were entitled to tuition reimbursement, and the Second Circuit agreed.