Jimmy Carter is said to be relaxed, in good humor, confident, even serene. That is the testimony of someone who saw him at length in the privacy of the White House family quarters a few days ago. If that description accurately portrays the president now, and at least one of his senior aides agrees with those characterizations, Carter's demeanor is remarkable indeed. For in many ways, the most difficult hours of his presidency appear to be approaching.

The Billy Carter business that has been dominating the news is damaging, but probably not destructive. It just happens to come at the worst of moments, and it rekindles old doubts about the Carter White House operation -- concerns that have more to do with personnel and personal relationships than with handling of policy issues. The danger for the president's reelection prospects lies in the timing.

At the period when the president seems to have hit bottom in the polls and would normally begin fighting back, Brother Billy consumes precious time and attention at all levels of the White House. Nor does this ugly episode promise to end swiftly; the longer it drags on, the more it cuts into the president's ability to turn events to his advantage.

Two weeks from now the Democrats convene in New York. Until not long ago, the convention could be seen providing Carter one of his most rewarding political moments. He had decisively beaten Edward Kennedy, only yesterday's overwhelming favorite of pollsters and press alike; he had, in Ronald Reagan, the one GOP candidate that in yesterday's wisdom was seen as most beatable; he had looked forward to doing what he does best -- taking his case to the people in a vigorous campaign.

Now all those positives look like negatives. He trails Reagan by some 28 percentage points in the polls, he can't shake the Kennedy challenge or bring about a desperately needed rapprochement with that critical Democratic wing, he faces growing discontent within party ranks that stirs new talk of a "dump Carter" movement, and he finds his campaign planning affected by the continuing albatross of Brother Billy's problems.

On top of all these downers, this weekend brings even more troubling news, starling in their implications and disastrous if their message correctly forecasts the political future.

Lou Harris, the pollster, has just received the results of his latest national survey. They show Republicans ahead in the races for the House of Representatives by 47 to 43 percent nationally.

The situation is even worse in eight of the largest industrial states -- Massachusetts, New York, New Jersey, Pennsylvania, Ohio, Illinois, Michigan and California. There, the Republicans are running ahead of the Democrats by 50 to 41 percent. These, of course include the states a presidential candidate has to win.

But beyond this, the Harris figures spell out something more potentially historic than an elected incumbent president losing the White House for the first time in nearly half a century. If the Republicans gain control of both the White House and the House of Representatives on one election day, it would be the first time they have accomplished that feat in over a generation. And it would signal a political change that goes far deeper than the failure of one president to help carry his party to victory nationally.

In effect, Jimmy Carter would be presiding over the disintegration of the Democratic Party.

Carter's weakness nationally is not new, it should be said. Four years ago, when the Democrats swept the Congress and recaptured the presidency, Carter ran behind the ticket in most areas. The coattails belonged to the members of Congress rather than the presidential candidate. It was they who helped pull him to victory.

Now, if Harris' latest surveys are right, Carter proves even more of a drag on the Democratic prospects. Reagan's present commanding lead over Carter helps sweep Republican congressional candidates along with him.

There is, in these findings, something of a political "smoking gun" syndrome at work. The Harris poll showing control of Congress in jeopardy is certain to generate greater pressure on Democratic convention delegates in New York to rethink their positions on going with Carter again.

Political death-knells come easy in the Washington summer when the news is slow, the press at ease, and the presidential election season drawing close. Any speculation about Jimmy Carter's demise, in particular, should be treated with extreme skepticism. He has proven a far more formidable, tenacious politician than he has ever been given credit for, as the political results this year have demonstrated so strongly.

Washington's least well-kept secret involves the search over the last six weeks or so for a Democratic alternative to Carter at the convention. The scenario, to use that favorite Washington operative's term, has been drafted out of dreams, animosities, wishful thinking and, yes, Carter's changing fortunes. But as yet not one major political figure has emerged to barnstorm the country, with the tacit blessings of the party's elders, to attempt to persuade Democrats nationally to throw open their convention and start fresh with someone else. Among Carter delegates, no significant defections have yet occurred.

Privately, some Democrats have evoked that old political cry -- now is the time for all good men to come to the aid of their party. No one has responded.

The belief in this corner, remains firm that Carter would show himself to be a much stronger candidate against Reagan than is now assumed and quite possibly victorious in November. But there's no doubt either that in the present hothouse, rumor-filled atmosphere of Washington, events on many fronts are generating a life form of their own. For Jimmy Carter, they pose the greatest peril of his presidency.