We support the military action, even if it is much belated, and the good news is that the first allied salvos from the air seem to have achieved initial success. They have knocked Gadhafi’s air force out of the battle and stopped his ground forces from advancing further into the rebel stronghold of Benghazi. Allied planes have also hit Gadhafi’s armor and troop columns, which ought to give his mercenaries in particular reason to ask if the pay is worth the risk.

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But the war’s early prosecution also raises concern about its leadership, its limited means and strategic goals. On none of these have coalition members been clear or unified, starting with President Obama.

It isn’t even clear who is commanding operation Odyssey Dawn. Admiral Mike Mullen, Chairman of the Joint Chiefs, wasn’t able to provide a clear answer as he worked the Sunday news circuit. Mr. Obama said on Saturday the U.S. will “contribute our unique capabilities at the front end of the mission”—presumably B-2 bombers and command and control—but he added that the no-fly zone “will be led by our international partners.”

Will that be the French, who said yesterday they have a handful of planes flying over Libya? It won’t be the Qatar air force, which is chipping in four fighters. It isn’t even clear whether the NATO commander will be allowed to lead the mission, though the military alliance is equipped for precisely this kind of effort. The danger here is that if no one is in charge, then no one is accountable for success or failure.

It also isn’t clear what the military and strategic goal of this operation really is. Reuters quoted Secretary of State Hillary Clinton as saying on Friday that the goal was “Number one: Stop the violence, and number two: We do believe that a final result of any negotiations would have to be the decision by Colonel Gadhafi to leave.”

Yet President Obama offered only the first aim in his statements on Friday and Saturday: “We are not going to use force to go beyond a well-defined goal—specifically, the protection of civilians in Libya.” He even suggested that if Gadhafi honors the U.N. demand for a cease fire, then the allies would stop fighting short of ousting him from Tripoli. On Sunday French Foreign Minister Alain Juppe explicitly rejected the goal of ousting Gadhafi.

A Danish F-16 aircraft flies over a Danish Royal Airforce C-130 as it takes off from the Nato airbase in Sigonella, on the southern Italian Sicily island.

Gadhafi is weak enough, and Libya is a puny enough military power, that even a limited use of force might lead to his ouster. Perhaps the officers around him will mutiny, though they would also have to defeat the Gadhafi sons who control their own mercenary bands and could be prosecuted for war crimes if they leave Libya.

Certainly Gadhafi showed no sign of retreat Sunday, promising “a long war” and revenge against the U.S., France and the United Kingdom. He already knows, thanks to the limits of U.N. resolution 1973, that he needn’t fear any foreign troops parachuting into Tripoli. He received further encouragement from Arab League chief Amr Moussa, who only a day into the allied bombing denounced civilian casualties and claimed this wasn’t the kind of no-fly zone the Arabs had in mind. Mr. Moussa is running to be president of Egypt, but U.S. military action should never be hostage to such a fair-weather ally.

The danger for the region, and U.S. interests, will be if Gadhafi can exploit divisions on the global war committee and achieve a military stalemate. He could then remain in control of a rump part of Libya and still create mayhem.

Even Admiral Mullen conceded that the war could end in a stalemate with Gadhafi staying in power. “Certainly, I recognize that’s a possibility,” he said on CBS’s “Face the Nation.” “It’s hard to know exactly how this turns out.” When America’s top uniformed officer says he doesn’t know what the goal of a military engagement is, you know he’s not getting clear direction from political figures in the U.S., or the global committee, or whoever is really in charge.

Mr. Obama’s own chief terrorism adviser, John Brennan, warned late last week that Gadhafi “has the penchant to do things of a very concerning nature,” including the possible use of his stockpiles of mustard gas. If Gadhafi poses such a threat, as we agree he does, then it is essential that this war end with a new government in Tripoli.

That means not agreeing to a premature cease fire that treats the opposition as no different from Gadhafi’s troops. It means aiding the rebels—with intelligence and other arms in addition to air cover—to rout Gadhafi’s forces. At the very least, the U.S. ought to recognize the National Council in Benghazi as a provisional Libyan government, which will enhance its international standing and ability to arm itself. We also see nothing in U.N. Resolution 1973 that would bar the U.S. from assisting the rebels with advisers as we helped Afghans topple the Taliban in 2001.

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The other problem with war by global committee is that it diminishes the role of the U.S. Congress. As he ran for President in 2008, Mr. Obama made much of his opposition, in contrast to Mrs. Clinton, to the 2002 Iraq war resolution in Congress. Yet so far regarding Libya he has been far more solicitous of the U.N., the Europeans and the Arab League than he has of domestic political consent.

We believe that, as Commander in Chief, Mr. Obama has the authority under the Constitution to order U.S. forces to act as he has in Libya. But as a simple prudential matter, a U.S. President needs to respect and bring along Congressional leaders in support of such action. All the more because members of his own party will be the first to revolt if a stalemate ensues or the TV pictures get ugly. Republicans tend to defer on principle to Presidential war decisions, but Mr. Obama also cannot afford to take them for granted.

The worst offense a Commander in Chief can make is to commit U.S. military force and the credibility that goes with it in half-hearted fashion. Now that he’s taken the U.S. to war against Libya, Mr. Obama needs to make American interests his main priority, and that means ensuring that the result includes a rapid end to the long, brutal rule of Moammar Gadhafi.