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The judges were left searching for limits on either side — and wondering why they were being asked to decide at all.

“Why drag us into it?” Judge Griffith asked the plaintiffs, hinting that the court could duck the entire issue, calling it a political controversy between two equal branches of government that should be solved by the political system.

The plaintiffs include Noel Canning, a bottling company that brought the lawsuit; the U.S. Chamber of Commerce; and Mr. McConnell and 41 other Senate Republicans.

While Mr. McConnell took a position in the case, the whole Senate — controlled by Democrats — did not. That fact troubled Judge Griffith, who said the court is left guessing at whether the Senate did, in fact, consider itself in session when the appointments were made.

Democrats began using pro forma sessions to block nominees under President George W. Bush, but stopped when Mr. Obama took office in 2009. When the GOP took control of the House in 2011, however, it used another part of the Constitution — which prevents either chamber from being away for more than three days while the other is in session — to force the Senate to stay in session, thus preventing recess appointments.

Victor K. Williams, an assistant professor at the Catholic University of America School of Law who has filed an amicus brief in the D.C. case supporting the administration, said the judges don’t have jurisdiction to hear the case because it is a political question.

“President Obama needs a new lawyer,” he said, questioning why Ms. Brinkmann, the administration’s attorney, never raised the jurisdiction issue with the court.

He argues that the framers intended for the president to be able to fill all positions so that the laws can be faithfully executed.

“Judge Sentelle and Judge Griffith have yet to go back to the summer of 1787 that they claim to revel in. If they were to go back to the summer of 1787, they would see the framers were intent on restricting the legislature’s appointment role,” he said. “They would see the recess appointment clause, as I’ve tried to frame it in my amicus brief, was the capstone of a summer’s work to try to restrict the legislature.”

He said presidents were intended to have a free hand to staff offices.

While Judges Griffith and Sentelle peppered attorneys with questions, Judge Karen L. Henderson stayed out of most of the back-and-forth. Legal analysts also caution that oral arguments don’t necessarily foreshadow final decisions.

The D.C. challenge is one of several proceeding through the courts.

Last week, the 7th U.S. Circuit heard another case, and those judges were less impressed with the challenge. One judge described the danger of a president using a Senate lunch break to make appointments as “pure fantasy.”

But the D.C. circuit judges seemed to take that more seriously.

Miguel Estrada, who represented Senate Republicans in the case, said that if the court accepted Mr. Obama’s interpretation here, it would open the door for the president to also declare filibusters or other informal Senate traditions to be violations of his ability to get an up-or-down answer on a nominee.