The prospects for that amendment seem far from clear. On the one hand, the more limited measure to restrict funding failed to pass. But on the other, the amendment attached to the appropriations bill could be more popular among those members who felt the financing resolution amounted to back-door authorization.

As the Libyan conflict has dragged on, there has been increasing hostility toward the Obama administration in the House among Democrats who oppose the war and many Republicans who cite constitutional issues over authorization for the military operations. Those lawmakers argue that such authorization is required by the 1973 War Powers Resolution.

Under that law, presidents must end unauthorized deployments 60 days after notifying Congress that they have begun. If what the United States military is doing in Libya constitutes “hostilities” — the administration argues that it does not — then that deadline passed on May 20.

The United States has handed the leadership of the air war in Libya over to NATO, and has largely played a supporting role, offering things like aerial refueling, surveillance, and signal jamming. But it has also conducted about 90 missile strikes from piloted aircraft and remotely operated drones, aimed at Libyan air defenses and ground forces, and United States aircraft have flown more than a quarter of the total sorties.

The total cost of United States operations is expected to top $1 billion by the end of September.

In an effort to avert a vote that would embarrass the administration, Secretary of State Hillary Rodham Clinton met with scores of House Democrats on Thursday afternoon to urge them to vote against the resolution limiting funds, insisting that the collective effort in Libya was close to ousting Colonel Qaddafi.

The votes on Friday followed two hours of fascinating and sometimes fierce debate that was unlike any other in this 112th Congress.

“We don’t have enough wars going on?” asked Representative Dennis J. Kucinich, Democrat of Ohio, on the House floor. “We have to wage war against another nation which did not attack us?”