When Allen Thomas comes to Washington, he does what every proper visitor does and heads straight for the Capitol.

But instead of piloting a Toyota full of squawling babes, Thomas peers out over 300 feet of barge deck from the wheelhouse of the Peggy S., a 112-foot tug. Riding full, he's pushing 2 1/4 million gallons of fuel up the Potomac River to make the nation's capital run.

"That's just about right, put her on the Capitol Building," said Thomas as he passed under the Woodrow Wilson Bridge. He swung the wooden ship's wheel with a practiced hand until the barge prow was centered on the illuminated seat of U.S. government.

It was dark on a glassy calm April morning. "Breaking day," said Thomas, gesturing toward Blue Plains, where the sky grew pale.

He and his son, Allen Jr., had stood silent vigils through the night, watching the shoreline twist as the river snaked past the lights at Cobb Island and Colonial Beach and the dark stretches at Maryland Point and Mathias Neck Point, where the bottom drops off to 119 feet, the deepest place in the Potomac.

They and two crew mates were running Steuart Transportation Co.'s oldest vessel, a 1938 Navy tug fitted in mahogany and brass and powered by 1,200 whisper-smooth diesel horses. Tonight she was pushing a load of aviation fuel from Piney Point near the mouth of the Potomac to the Andrews Air Force Base pipeline near South Capitol Street Bridge. The 100-mile trip took 14 hours.

"You watch," said chief engineer John Tanner, "in a couple days President Reagan will take off somewhere in Air Force One. That's what they want this stuff for."

Normally, Peggy S. would be pushing a less romantic, less explosive cargo. Her standard load is No. 2 heating oil. In the cool and cold months she and Steuart's three other inland tugs constantly ply the Potomac, the Delaware River and Chesapeake Bay, bringing fuel from the company's 6-million-barrel storage facility in Piney Point to cities where it's needed.

Heading to Baltimore, Trenton, Wilmington or Philadelphia, the Steuart fleet joins a stream of commercial traffic spirited along by sophisticated navigation aids. The way to Washington is lonelier.

Steuart is the principal commercial user of the Potomac, where there is little industry to support other navigation. Bound for a city where hot air and paper are the major products, Steuart boats only rarely pass a freighter loaded with newsprint for Robinson Terminal in Alexandria or another barge loaded with heating oil.

It makes for interesting voyages. On the better-traveled rivers, said Thomas, "we just follow the range lights on shore." On the Potomac there are none, so he peers all night into the radar scope, picking out fishing nets along shore and buoys marking the channel.

There are surprises. All the lights on the Rte. 301 Bridge at Morgantown were out tonight, so he found his way under the gloomy arches by radar and the brilliant spotlight on the cabin-top.

There is no room for error. Sand shoals build where the river bends and there are bends aplenty. At Kettle Bottom Shoals, two hours out of Piney Point, the channel narrows to 75 yards wide in a river three miles across. There is only one small safe place for a barge drawing 17 feet. Thomas dances from scope to spotlight to helm, keeping on course. "You hit with a load of aviation fuel, you're gone," said his youngest son, James, the fourth man on Peggy S.

Arriving in Washington by churning tug lends new mystery to this mysterious city. There is no rosy dawn; it's rosy all night. A street light halo hovers over the city, visible as far downstream as Indian Head. But for this veteran crew it's just another city and just work, better only because it's nearly done. There is a calendar in the galley where the days are marked off with meticulous Xs, and the calendar says two days remain on this tour.

Whatever port they're in in two days, relief will arrive. Tanner's father will fly in from Portsmouth, Va., to take over the engineering from his son. A relief captain and crew will turn-to.

Steuart will fly the Thomases home to Tangier Island, Va., where Allen Sr. will keep busy running for mayor. The men work two weeks on, two off.

"That's what's nice about tugboating," said Allen Thomas Jr., "them two weeks off. When you come home to your wife it's like a honeymoon all over again." He laughs. "For about two days, anyway."

The men stand six-hour watches, which puts Allen Jr. at the helm from midnight to 6 a.m. But tonight, after passing Mount Vernon at 4, he snuffs out a cigarette and calls his father from his bunk.

"Fort Washington, dad."

The elder Thomas rolls out and takes the helm. The river streams by, turbulent and coffee brown in the deck lights.

"Right around the bend you'll see Wilson Bridge," says the captain.

And beyond it the Capitol, shining white across the slick water, the last landmark.