There is a saying among riders: "You have to throw your heart over the fence, and all you can do is follow."

This year, Melanie Smith, 28, has followed her heart over more fences more successfully than any other American rider. At the 20th annual Washington International Horse Show (at the Capital Centre through Sunday) some horses sail over the barriers, some lunge and others seem to glide. When Melanie Smith is riding, the horses jump. Her approach is athletic, workmanlike - not always the most graceful, but highly effective.

So effective that, for the first time, whe has jumped the biggest hurdle of them all - after years of riding as an individual, she is a member of the U.S. Equestrian Team.

Also on the team is her spectacular partner, Val de Loire, a 13-year-old French-bred gelding, who she met last year in Europe and persuaded her sponsors to buy for her. "He was on the Spanish team and they needed money, so we got a very good price," she says. Rumor at the show puts the price at about $50,000. His current value is estimated at four to five times that much.

"he is just a wonderful athlete, a talented jumper, a powerful horse," Melanie says. "He has a unique personality. He's a fun guy and loves attention, but he's also quiet and sensitive - just a very good friend."

To anyone who knows "National Velvet," riders like Melanie (merging vaguely into the figure of the young Elizabeth Taylor - or, more recently, Tatum O'Neal) are a special sort of heroine. But it is not all romance and dreams and love of horses in serious competition.

For one thing, it costs Melanie's sponsors - Mr. and Mrs. Neil Eustace of Stonington, Conn - well over $100,000 a year to send her around the circuit.

Among horsemen, a common estimate is that it costs $15,000 per horse to get through a season from Elorida in February to Madison Square Garden in November.

This includes food and lodgings for the animals and humans, as well as entry fees and transporation costs, but it does not include the basic capital investment. The horse themselves can cost anywhere from $10,000 to $100,000 each, or more - sometimes according to rumor, as much as $300,000. It is probably fair to estimate that Melanie, wit height to 10 animals, is accompanied on her tour by nearly $1 million worth of horseflesh.

Unlike racing, the owners do not invest this kind of money with their eyes on the purses. Idle Dice, who has been a champion jumper for 10 years and is still competing at age 15, has the largest earning in the history of the sport. Horse-show people point to him proudly as havingwon $250,000 - which means that he may have broken even financially. When asked why they do it, horse people have a stock reply: "For a cheap piece forayon - preferably blue."

For Melanie Smith, it all started back in Germantown, Tenn., a Memphis suburb, when she was 3 years old and her grandfather gave her a Shetland pony. "But within two years," she recalls "there were so many ponies around the place that we had to move out of town to a farm."

She competed for years in local shows ("I didn't use a saddle until I was 12 - just rode bareback"), and did not have formal training until she was 19.

Before then, riding in the midwestern and southern ciruits, she did not concentrate on jumping. "They do that Back East," she says, "and it took me a long time to realize that that's where the action is."

The action turned out to be in New Jersey with George Morris, a former member of the U.S. Equestrian Team, whom she calls, "the greatest trainer in the world."

"It took me two years to persuade him to take me on," she recalls. At 17, "I was too old, and he didn't have a place for me. When he finally agreed, I took care of the horses and worked as a groom to pay for my lessons."

She bagan jumping competition in 1970 in the amateur-owner division, and two years later won the Eisenhower Cup at the Washington International Horse Show - one of the most coveted trophies in the field.

"She was definitely a long shot at that time," recalls Vicky Moon, a friend who was riding the circuit with here at that time. "When she won it, everyone was in tears, including Melanie and the people who were competing against her. People are friendly on the circuit; they wish one another luck. Some girls share rooms and then go out and compete with one another."

"Girls" are everywhere on the horse-show scene - on of the few sports in which men and women compete on a basis of approximate equality, at least in theory. In practice, girls predominate in the junior and amateur categories, but when they become women - in competition at the national level, men predominate.

"Everyone talks about it, but nobody can explain it," says one woman who retired from riding a few years ago. "Some of the girls get married and drop out - you wonder where all the men came from all of a sudden. There aren't that many of them at the lower levels."

One place where men seem to predominate is the U.S. Equestrian Team. All Week at the Capital Centre, Smith's fans (who are numerous and vocal) could be heard grumbling about how late she eas added to the team - after its European tour this summer. Some suggest that the fact she is female had something to do with it.

"It's better if I don't talk much about that," Smith said when asked about it. "I don't know how team members are chosen - there probably should be a point system, so people could earn their way onto the team."

In fact, nobody seems to know how members are chosen, or even who can be considered a member of the team. Without an established roster like the Redskins, the U.S. Equestrian Team is a rather amorphous pool of qualified riders from which four-member teams are chosen for each specific event. Nobody knows yet who will be on the team that goes to Toronto in about two weeks; Smith is proceeding on the assumption that she won't.

"Sex isn't necessarily the determining factor," says Robert Ridland, who went to Europe with the team this summer. "Once in the '60s there was an all-female team sent to Europe."

In a way, being passed over for the team during the summer contributed to Smith's success. She stayed at home and competed and won while her major competitors were in Europe with the team.

After next week's show at Madison Square Garden, the climax of the U.S. horse show season, Smith and her horses will have their longest rest of the year - two month's in Stonington. "Rest" means getting up at 7:30 and riding six to nine horses per day, schooling young horses and exercising the show veterans on the flat: "I don't jump the competitors. They already know how to jump and they can do just so many jumps in one lifetime."

In January, Smith and the horses go to Florida for a month of acclimatization before plunging into that state's 10-week circuit. On the road, she is up at 6 every morning, exercising the horses, pacing the courses and planning jump strategy, competing in at least wo events daily, living out of suitcases and polising her boots. ("I hate polishing boots more than anything, but I do it every time before I show.")

Along with the eight or 10 horses, three people accompany her on the road to help care for them. During the 10-months season, they get back to Connecticut once in a while, but usually not for more than two weeks at a time.

Because of the intense concentration on her profession and the foot-loose show circuit, Smith says, she has little time for men in her schedule. "There's nothing serious in my life at the moment - just a lot of good friends. I haven't planned it that way. It just happened."

The life of a traveling equestrian is "like a gypsy," and would not fit in well with any kind of domestic routine, she adds. "It would probably have to be someone else who travels the circuit, and that cuts down the field."

But right now, she seems completely unconcerned about men, and such more interested in the happy end to her long, hard struggle, which is now in sight.

"Every rider's dream," she said, "is to ride with the U.S. Equestrian Team and to do it in Madison Square Garden."