The overhaul of the Senate’s filibuster rules was prompted, surprisingly, by an argument about judicial productivity. Republicans said President Obama’s nominees to three open seats on a federal appeals court in Washington were not needed because the court does not have enough to do. But trying to quantify the workload of the court, the United States Court of Appeals for the District of Columbia Circuit, “is a statistical food fight,” said Russell Wheeler, an expert on the federal courts at the Brookings Institution. The real issue, he said, was “the ideological balance of the court.” The court’s eight current active judges are evenly divided between Republican and Democratic appointees. If Mr. Obama’s pending appointments are confirmed under the new rules, which end the minority party’s ability to filibuster most presidential nominees, that balance will change. But the court also has six semiretired judges, five of whom were appointed by Republican presidents. Their workloads vary, but on average they collectively handle the work of roughly three active judges, according to a letter in July from Chief Judge Merrick B. Garland to the Senate Judiciary Committee.