Israel's display of anger at President Carter directly results from top-secret American replies to questions by King Hussein of Jordan about the future of Jewish settlements on the West Bank.

Those answers, published here for the first time, hint that President Carter expects Israeli inhabitants of the settlements to start clearing out of Arab territory at the end of the five-year transitional period. During that period, Israel, Jordan, Egypt and the Palestinians are pledged to restore "the legitimate rights of the Palestinian people," consistent with Israeli security.

In answer to the King's question about "the status" of Israeli settlers - "Will there by any [settlements] after the end" of the five-year transition? - the U.S. replied as follows:

"Whatever number that might remain beyond the transitional period . . . would presumably be agreed to in the negotiations concerning the final status of the West Bank and Gaza."

Those words infuriated the Israeli government, and particularly Prime Minister Menachem Begin, when the secret U.S. answers to Hussein were shown to the Israeli government on Oct. 19 and discussed with Begin by Assistant Secretary of State Harold Saunders the next day. To demonstrate Israel's defiance of President Carter for implying to Hussein and the Palestinians that the settlements will become vestigial remnants, Begin went public with his long-nourished plan to "thicken" the settlements.

Carter's response was equally angry. He took Secretary of State Cyrus Vance's draft of a presidential protest letter to be sent to Begin "and doubled it in spades" before cabling it to Begin.

Thus the settlements question, on which Carter feels he has been betrayed more than once by Begin, returned to haunt his Mideast peace hopes. Equally forbidding is this suspicion known to be held by the president: that Begin's settlements offensive stemmed less from demands by his hard-line Likud coalition than from his strategy to throw up barricades to prevent Hussein and the Palestinians from joining the peace talks.

Begin's fury at the U.S. answers to Hussein's 14 questions about the Camp. David agreement go beyond the settlements issue. For example, Hussein asked: "What is the U.S. position "on whether the self-governing authority to be created on the West Bank includes East Jerusalem "both in terms of territory and people?"

The U.S. will support proposals," said the U.S. response, "that would permit Arab inhabitants of East Jerusalem who are not Israeli citizens" to vote in the election leading to self-rule. Further, those Jerusalem Palestinians could share "in the work of the self-governing authority itself."

There is nothing startling in this statement of American policy on occupied East Jerusalem. Ever since the 1967 war, the United States has denied Israel's legal power to absorb East Jerusalem. The United States has never recognized Jerusalem as Israel's capital.

But the political impact of the U.S. answers to Hussein is linked in Begin's mind to Israel's magnanimous agreement to withdraw from all its settlements in the Egyptian Sinai. To Israel, the clear intent of the answers to Hussein is to encourage Arab hopes that the West Bank settlements are also doomed.

Israel's counterattack raised the threat of delay in the negotiations for a peace treaty between Israel and Egypt, despite the coincidence of a Nobel peace prize for both Begin and President Anwar Sadat. Beyond that, the Carter-Begin shootout risks future U.S. retaliation against Israel, which is seeking billions of dollars from the shrinking U.S. budget.

"Begin is slapping Carter in the face with one hand and reaching for carter's wallet with the other," one administration official said privately. He predicated the aroused president might cut "a cool billion" from the bill Israel is now preparing for the United States to finance the Israel-Egyptian treaty.

Such passions will cool. What will not change is Carter's Camp David pledge to be a "full partner" in the peace game. That was Hussein's first question. The United States replied that it would use its "full influence" to get a West Bank settlement and that "President Carter will continue to take an active personal part in the negotiations."

If Israel really is trying to scuttle the West Bank talks and get a separate peace with Egypt, as the White House suspects, those words brought no comfort to Prime Minister Begin.