The Harlem Globetrotters are looking for the perfect woman.

In the last 59 years, they have won 15,843 basketball games and lost 331 without the benefit of womankind. But now the call has gone out. The Globetrotters will be a coed team next year.

All over America young women who can go to their left are whistling "Sweet Georgia Brown." (The kind of woman who can go to her left knows without being told that "Sweet Georgia Brown" is the Globetrotters' theme song.)

There was no music and none of the usual comedy at the National Museum of American History yesterday. The Globetrotters arrived quietly on their red, white and blue custom bus, on their way from Philadelphia to Norfolk. They stayed just long enough to donate a uniform, a warm-up suit and all manner of memorabilia to a new exhibit called "A Constellation of Black Sports Stars" and to certify their place in history.

On display, behind the glass, were Jimmy Brown's practice jersey, Franco Harris' helmet, George Foreman's boxing shorts, Wilma Rudolph's bronzed spikes. On display, in front of the glass, were the Globetrotters. They posed. They frolicked. They rolled basketballs up and down their arms.

"Oooh, I touched Curly," said Kelvin Chapman, 14, of Landover.

Fred (Curly) Neal said, "I am excited about having a woman," he said. "Give her a chance. I welcome her with open arms. Of course, it might be a different thing as far as the bus rides are concerned."

"It's going to be kind of rough," said Hubert (Geese) Ausbie, 45, the elder statesman of the team. "But I can cope. Traveling on the bus, we don't have no home town. The wear and tear on the body, I've seen guys get tired. For a woman, I don't know. We play some big guys."

"We started talking about it after the summer, after the Olympics," said Stuart Zanville, marketing director of the Globetrotters. "A lot of people say the Harlem Globetrotters no longer provide things people cannot see anywhere else. Michael Jordan [of the Chicago Bulls] can do things our players can't do."

So they will do what the NBA can't or won't.

They have received 100 to 200 applications from women and hope to narrow them down to five or six before next summer's tryout camp. Size is not the most important criterion; a radiant personality is.

Zanville said they have contacted many of the top names in women's basketball, black and white: Lynette Woodard, the leading scorer in women's basketball history; Pam and Paula McGee, twin forwards from the University of Southern California; Nancy Lieberman, the red-headed Olympian from Queens, N.Y.; and Ann Meyers, the only woman ever to attend an NBA camp.

Woodard, who played for the 1984 gold-medal Olympic team, is now an assistant coach at her alma mater, Kansas University.

"As you know, the Globetrotters are known for their wizardry with a basketball," she said over the telephone. "I'm working on that." Once when she was 6 or so, her cousin Geese Ausbie came to dinner. "From that point on, every time I heard 'Globetrotters' I lit up inside," she said.

"It's time to see if one would fit in," said coach Charles (Tex) Harrison. "Since we were the first to do a lot of things, we should be the first to do this."

"It may be more than one woman," Zanville said. "It's going to be lonely out there."