To understand the dynamics of the Middle East arms race -- and its essential mindlessness, as well -- you have only to put this question to almost any defense expert, in or out of office:

Why did the United States originally agree to sell 60 of its most sophisticated military aircraft, the F15 fighter-bomber, to Saudi Arabia, and why is it now ready to significantly enhance both the range and firepower of this plane by adding fuel tanks, advanced air-to-air Sidewinder missiles and other support aircraft and equipment?

The answer: Because the Saudie want them.

Next question: Why do the Saudis need this capacity to strike at targets 1,000 miles away? The answer, almost unanimously, is that they don't. So why, once again, are we doing it? At this point the conversation comes full circle: The answer is that the Saudis are insisting on it.

Strictly speaking, of course, that's only part of the explanation. The other part is that the United States, for reasons that make almost no sense in military terms, can't bring itself to say "no" to Saudi Arabia.

Now it is true that along the way, in the case of the F15s, the United States has said no. President Carter explicitly promised not to sell the fuel tanks, missiles and all the rest when he was fighting a close and bruising battle with Congress for approval of the sale of the aircraft in 1978. Israel and the American Jewish community were violently opposed.

And then again, when the Saudis began pressing for the "extras" last year, some 68 senators expressed their opposition in a letter to the White House. That would have been more than enough to kill the sales under a procedure that permits either house of Congress 30 days to veto such transactions. Just 10 days before last Noverber's election, in what was obviously an effort to reassure Israeli supporters, Carter publicly promised "no change" in his original commitment not to upgrade the F15s' capabilities.

One is left with the clear impression that the Reagan administration is now reversing Carter policy -- that this is a partisan matter, reflecting the new administration's greatly enlarged concern for Mideast, and Persian Gulf, security, and consequent willingness to accede to Saudi demands in a way that the Carter administration would not.

That's what the public record suggests. But the private record suggests quite the contrary. According to former high officials in a position to know, almost immediately after the election, the Saudis once again began pressing their case for the extra gear for the F15s with renewed urgency. And the Carter administration was ready, in the middle of the presidential transition, to go along, at least most of the way.

Quiet overtures, in fact, were made to the Reagan transition team. An offer was made to make it a bipartizan affair, with the president-elect giving his tacit approval in a way that would have taken both the outgoing and the incoming administration off the political hook.

But the Reagan camp reportedly was not yet ready to take the plunge, although there was strong support among Reagan's military and foreign policy advisors for upgrading the F15s. The Carter administration thereupon decided against going the F15s. The Carter administration thereupon decided against going it along.

"We never did make a binding commitment to the Saudis," says one former Carter official. "But the written record if published, would certainly imply that we were ready to go forward. If we had been reelected, we would probably be doing pretty much what the Reagan administration is doing now."

The effect of all this is to make the F15s pretty much of a non-issue. What seemed to be shaping up as another battle royal, with Democratic Sens. Kennedy, Cranston and Biden leading the opposition, is likely to be not much of a fight.

The Reagan administration is putting it about, in its dealing with American Jewish leaders, that it is only doing what the Carter administration was prepared to do. The Israelis, sensing the force of that argument and not being eager for a losing confrontation with the new American administration, are registering only perfunctory opposition.

One reason: As an offset to the supposedly new military capacity being conferred upon Saudi Arabia, Israel will receive some $600 million in new arms credits from the United States.

And so the Mideast arms race spirals on, with no clear answer as to how the additional arms fit into U.S. strategic planning to meet what the Reagan administration perceives to be an increasingly dangerous Soviet threat to the area.

There are undoubted Saudi defense needs -- for defensive weapons such as the far less sophisticated and more immediately useful F5s, which the United States has furnished Egypt. The question remains what use Saudi Arabia has for the hottest and hardest-to-handle offensive aircraft in the American arsenal. If it is a matter of prestige -- of ingratiating a vital oil producter, one with a potentially constructive role to play in the Palestinian issue -- a further question remains. What has the United States so far received -- or can reasonably expect to receive -- in return?