JIMMY CARTER has now made a ringing and detailed defense of his integrity in the Billy affair. His evident intent, besides clearing his name, is to clear the air before the Democratic convention and to preempt the Senate investigation.Has he succeeded? Tune in in a week or two. Our guess is that, when the poking in corners and the holding up to the light have been completed, the story will have faded as an event but will be flourishing as a symbol of people's complaints about the administration. The obvious innocence of the president in using his brother to reach the Libyans so that they could be urged to needle the Iranians on the hostage question seems to us to wash our what suggestion of illegalities or improprieties has been made so far.

But the president still does not acknowledge a troubling aspect of this episode. It is not his integrity but his judgement that is in question. He makes out that his "bad judgement," in using Billy Carter as a conduit though he knew that would build up his brother, was actually a concealed virtue, good judgement : "I did what I thought was best for the country and best for the hostages." But in our view the real bad judgement lay in his use of a brother to make the contact when the need for using him was at best questionable and when the use of him gave the Libyans, or so they seem to have thought, influence of a sort they could not otherwise have dreamed of.

Consider that the State Department, on Nov. 8 and 18, had gotten positive responses from quiet probes meant to get the Libyans to go on the record as being against the hostage seizure of Nov. 4. At White House suggestion, Billy Carter got in touch with a Libyan diplomat here on Nov. 20. On Nov. 22, the Libyan government came out against holding hostages.Billy Carter brought the Libyan diplomat to meet the president's national security adviser on Nov. 27, and Col. Qaddafi, the strongman, sent Mr. Carter a message on Nov. 29. Libya burned out the American Embassy on Dec. 2. To discuss that as well as the hostages, the president himself met with the Libyan diplomat on Dec. 6. On Dec. 11, Col. Qaddafi said in a newspaper interview that he was suspending earlier threats to curtail oil exports since he had received "assurances in the last few days through unofficial but reliable channels from President Carter" that, if reelected, Mr. Carter would take "a more neutral American posture in the conflict between the Arabs and Israel . . . a more sympathetic attitude toward the Palestinian people."

What actually happened? On the one hand, Mr. Carter, with "an absolute, total obsession" with the hostages, did not want to leave the new crisis, any more than any other aspect of Iran policy, to the State Department. On the other hand, his bent for family diplomacy was strong. So through his brother and the White House staff, he reached out to the Libyan diplomat and opened up an exchange with Col. Qaddafi. His personal notes make clear he was gratified that Libya was cooperating on the hostages and hopeful that Libya would make amends for the embassy burning. He looked ahead to "better relations" with Col. Qaddafi, hereto-fore known to him chiefly as a sponsor of terror, armed insurrection and anti-Camp David trouble. Over the whole sequence hung his relief that Libya was continuing to pump lots of oil.

To judge by Col. Qaddafi's press interviews of Dec. 11 and Jan. 10, he was delighted by the new turn. The White House was asking him merely for the help on the hostages that he had already started offering in response to the State Department Jimmy Carter was thanking him on the hostages even after the embassy had been burned. Best of all, he had shed his semi-out-law status for a direct channel to President Carter, who now was talking of "better relations." These were the conditions in which Col. Qaddafi began speaking publicly, over White House denials, of White House "assurances" of an American Mideast policy change in 1981. Is it a surprise that the Libyans, having paid Billy Carter nothing for his previous public relations chores, now gave him $20,000 and then $20,000 out of a promised $50,000? Who was using whom?

President Carter now proposes to write a rule barring executive branch employees from dealing with any member of his family "under circumstances that create either the reality or the appearance of improper favor or influence." That puts the onus on his bureaucrats. It should be on the officeholder whose family is the one in question. At the core of this episode is Mr. Carter's belief that members of his family are often his best surrogates -- that he shows respect to foreigners and personal interest and commitment by having them deal with persons related to him.

But this is wrong. It is not Mr. Carter (or any president) the man, but rather Mr. Carter the president, with whom foreigners are dealing. He represents the government in his involvements with them. He can honor them and show them that he takes them seriously by sending other high elected or appointed officials to represent him rather than members of his family. A president's family has no standing on our government or system. Public officials do. Mr. Carter's attitude, which is the reverse of this, invites precisely the confusion that has put the Billy affair at the debilitating center of the administration's concern. The way to avoid family-bureaucrat dealings that create the reality or appearance of improper favor of influence -- and contacts, like the Billy-Libya one, that hurt the president and his capacity to govern -- is to avoid family-bureaucrat contacts. The way to avoid those contacts is to terminate them from the family end. If any new rule on such contacts is needed, it is a self-denying ordinance that Mr. Carter should write for himself.