Washington ? Is the vice president losing his influence, or perhaps his mind? That question, even if it is phrased more delicately, is creeping through foreign ministries and presidential offices abroad and has become a factor in the Bush administration’s relations with the world.

“What has happened to Dick Cheney?” That solicitous but direct question comes from a European statesman who has known the vice president for many years. He put it to me a few days ago – even before the discovery of a blood clot in Cheney’s leg and the perjury conviction of Scooter Libby, his former chief of staff, brought headline attention to the volatile state of the vice president’s physical, emotional and political health.

It is not new for Americans to question whether their leaders have become delusional. Editors at The Washington Post directed reporters to find out if Jimmy Carter had suffered a nervous breakdown when he retreated to Camp David in 1979 for 10 days and abruptly fired five Cabinet officers. Remember the hubbub over Al Haig’s “I am in control here” and other Captain Queegish remarks, or Richard Nixon’s talking to portraits?

What is unusual is for foreigners to think about a vice president at all and to question what effect the VP’s moods and internal policy defeats have on America’s standing in the world.

But what goes up must come down. In the first term, Cheney was styled as the most influential vice president in history – in more lurid versions, an evil puppeteer pulling George W. Bush’s strings. So now his irascibility in television interviews triggers diplomatic cables analyzing his equilibrium – as well as inspiring a booming industry of scathing cartoons and television one-liners here at home.

And it is not over. Reports of a new defeat lie ahead for the hard line on Iran and Syria that is associated with Cheney’s office if this week’s meeting of ambassadors in Baghdad produces progress on Iraqi stability. Diplomats tell me that Secretary of State Condoleezza Rice has persuaded Turkey to host a ministerial conference next month that will include Iraq’s neighbors, the permanent members of the U.N. Security Council and G-8 nations, and that she will attend.

Rice is credited by administration sources with having told Bush in January that he should devote his final two years in office to seeking diplomatic agreements with North Korea and Iran and an Israeli-Palestinian peace accord. That account emphasizes that Rice is not simply outflanking Cheney in intermittent internal policy battles, but has won full agreement and support from the president on the strategic goals and methods she and her diplomats are pursuing.

This remains to be confirmed by events. But it is clear that Bush has always been much more the decision-maker than the Cheney-as-puppeteer image conveyed. It is not just recently that Bush has failed to follow Cheney’s counsel:

On Iraq, Bush overruled Cheney on going to the United Nations for a second use-of-force resolution and then listened much more closely (and disastrously) to policy prescriptions from proconsul L. Paul Bremer III and others. On Iran, Cheney came to office with relatively more relaxed attitudes than Bush. Cheney’s attitudes may have been formed by his experiences as chief executive officer of Halliburton, an oil services company that has sought out business contacts with that nation.

There is much to credit in Cheney’s frequent protestations that unmitigated loyalty to Bush is more important to their relationship than the policy advice that Cheney gives the president. It is advice that he never discloses to associates in the Cabinet or the few diplomats whom he sees.

“It is so mysterious that his recent public hints at discomfort with the new policy directions reverberate with us like muffled cries of outrage,” says one ambassador here.

The Libby trial revealed serious splits between Cheney and Bush’s political team led by Karl Rove, who suffered no legal consequences for his role in the scandal. The trial also served as another exercise in showing how Cheney has empowered his critics at home and foes abroad: His excessive concern for secrecy and control by the executive branch has given new credibility and fundraising ability to the Democrats and to civil liberties organizations here, and has won sympathy around the world for prisoners who may well be terrorists.

So listen up, diplomats: However beleaguered, Cheney will not resign over the president’s refusal to take his advice. The only force that could drive him to that dramatic step would be that unshakeable sense of loyalty to Bush, who desperately now needs a vice president in stable physical, emotional and political health. That is the equation you want to be watching.