Representatives of airlines seeking to operate out of Washington's National Airport this summer grappled all day today without making any headway in the task of reducing their requests for landing privileges at the restricted airport.

Without a massive change in circumstances, the problem of dividing access to National among the airlines again appears destined to be put in the lap of the Department of Transportation.

The DOT was called on last year to allocate landings and takeoffs -- called slots -- at National when the airlines themselves were unable to agree on a formula for the current airline scheduling season, which runs through April 25. The allocation task is even more formidable for the summer season, which runs from April 26, the day daylight savings time begins, through the day in October when it ends.

Under a DOT plan scheduled to take effect with the new season, the number of slots allocated to commercial airlines flying airplanes with 56 seats or more will be reduced almost 20 percent, from the current 640 a day to 522. Besides reducing the number of operations from 40 to 36 an hour, the plan also imposes a 9:30 p.m. curfew and introduces widebody jets at National. (The other four slots per hour are added to those available to the commuter airlines).

The meeting today started with requests from 24 airlines -- up one from the number currently operating at National -- for a total of 812 slots a day. However, because 6 of the airlines said they will be using planes with fewer than 56 seats, their requests for 60 slots were removed from this airline scheduling committee's deliberations.

That leaves 18 airlines seeking 752 landings and takeoffs a day when only 522 are available.

In a day marked by acrimonious remarks, the airline representatives also failed to come to any agreement on how to redistribute some current unused landing and takeoff privileges. Representatives of Braniff International and Pan American World Airways both said they would not participate in a drawing for the slots unless they each got one first. Both airlines were among a dozen that were forced to give up or move flights to 10 p.m. when they were divided under the DOT-formulated plan. Since then, however, several airlines have given up some slots voluntarily, making them available to other airlines. Eleven airlines originally were seeking some of the unused slots. "It's a crying shame . . .," Nestor N. Pylypec, chairman of the committee, said.

Today's meeting was the second this week on airlines' proposed schedules at the four U.S. airports that are restricted -- National, New York's LaGuardia and John F. Kennedy International Airports and Chicago's O'Hare International. JFK and O'Hare restrict takeoffs and landings from 3 p.m. to 8 p.m. daily.

On Monday, in a room without windows at a beachside hotel here, representatives of 19 airlines seeking operations this summer at LaGuardia grappled all day without any progress to fit requests for 870 slots into an airport limit of 768 operations a day.

The task of the airline scheduling committees has grown almost impossible as airline deregulation has spawned new airlines and has given existing airlines easier access to new routes. Twenty-three airlines are using the valuable slots at National now compared with 10 two years ago.

The airline officials both days have been reluctant to agree to lower their requests both for competitive reasons and apparently out of fear that if the DOT is called in to settle the requests, it will use as a base the requested number of operations.