Elizabeth Huffman, who is a little girl, wrote a note six year ago and taped it to her bedroom window so that it could be read from the heavens. It said, "Please God, bring me a horse."

Cathy Reckert, who is not a little girl, owns a thoroughbred horse. She loves to ride horses, talk horses and watch horses. This week she bought a new $359, 19-inch color television set to watch "Black Beauty."

Dick Spencer, who is an airline pilot, has a wife who owns a horse named "I'm A Witted." His wife Nancy calls the horse "Witted" for short. She says, "He (the horse) loves me." Her husband calls the horse "Half-witted" and he does not speak of love.

Elizabeth Huffman, whose request for a horse was answered last October, Cathy Reckert, and Nancy Spencer are members of the nouveau horsey set in Fairfax County. Because Fairfax has more than doubled in population since 1960 to about 570,000 people and because much of the growth has been scattered with enough room between houses for horses, horse experts say the county is one of the hot spots in the nation for the development of an ever-growing breed - the suburban horse.

"If you move to the country, you inevitably want a dog, a vegetable garden and a horse," says Jacqueline Mock, co-owner of Potomac Equitation Farm in Centreville.

The horsey set in Fairfax differs considerably from the horsey set in Loudoun or Fauquier counties or in the Potomac area, according to some whose business it is to know horse owners.

Peter Winants, editor of the nation's largest horse weekly, "The Chronicle of the Horse," says horse owners in Fairfax generally have cheaper horses, less horse knowledge and far less money than the landed gentry farther away from Washington. As one Fairfax horse owner puts it, "We are tawdry commoners compared to people who have their stable hands set aside a day each year to polish silver stirrups."

There are about 4,400 of these "tawdry commoners" riding about 5,000 horses and spending about $5 million a year in the county for the privilege, according to a recently completed survey of horse owners in three of the county's largest horse-owing districts, Dranesville, Centreville and Springfield.

The survey, conducted by county horse owners with the technical assistance of the county office of statistics and research, says horse owners support some 61 business in the county.

"In this area the horse population is so great and so concentrated and horse owners' income is so high that you can make a lot of money if you know how to run a business," says John O'Hare, co-owner of Dominion Saddlery near Chantilly.

Large animal veterinarians, who have to treat a horse three or four times a year at the minimum, say they make between $60,000 and $80,000 a year. Blacksmiths, who trim a horse's hoofs every six weeks, say they make around $25,000 a year. Feed store, tack (horse supply) shop and stable (owners won't say how much they make, but they smile when asked if business is going well.

The source of the bounty, horse people say, is the little girl who prays and pleads for a horse and the not-so-little girl who always wanted one.

"When the little girls come into my store," says O'Hare at Dominion Saddlery, "it's like they were let loose in their own little paradise. They just don't want to leave."

If a little girl has just been given a pony (which sells for between $1,000 and $2,000 if it is trained not to bite and kick) she can easily spend $500 of her parents' money out-fitting her horse and herself before she leaves the "little paradise."

Those who make their living off horses owned by little girls and women (which accounts, they say, for about 85 percent of the horses in the county, estimate horse owners come from upper-middle-class families with incomes averaging $50,000 a year.

The nouveau horse people from the families of bureaucrats, military men, doctors, lawyers and business men in the county are easy to spot, according to the few "Old Virginia" horse people who remain in Fairfax County.

Jane Marshall Dillon, who runs Junior Equitation School near Vienna and is a fifth-generation descendant of famed U.S. Chief Justice John Marshall, says people new to horses in the county work hard to learn about their animals.But she says their taste in horses is, at best, unrefined.

"Many of us who've been exposed to horses all our lives almost find it hard to understand or tolerate the woman who says about just any horse, 'Oh, what a beautiful horse.'"

Nancy Spencer, who bought the first horse she's ever owned a year-and-a-half ago, says she paid $1,500 for a tempermental 14-year-old thorough-bred because she thought he was "beautiful."

She and her husband Dick and their two children moved into a house with seven acres near Clifton in the sparsely populated western part of the county three years ago. She immediately started thinking about a horse.

"Everyone on our street had a horse," Mrs. Spencer says, "so I enrolled the kids in riding lessons." She went to every one of the children's lessons, then started taking lessons herself.

"She had this burning desire to get her own horse," says her husband, an American Airlines pilot who enjoys running and dislikes horses. "Then she got one and it's gone down hill from there."

The Spencers admit that their horse, "Witted," which costs about $2,500 a year to ride and maintain, is not exactly the kind of horse they want. If is skittish, eats a lot, needs frequent hoof care, is hard to ride and has one running speed - fast.

It is common for the suburban horse to be the wrong horse for its owners, says Mock of Potomac Equitation Farm.

She cites the case of one county family that bought for their son an undernourished pony with bad temperament, that was difficult to saddle or ride. The pony kicked the boy in the mouth; oral surgery was required. The family's veterinarian told them, "Green (horse owners) and green (ponies) don't mix."

Another mixture that does not work in the world of the suburban horse is boyfriends and horses. Lisa Spencer, 15, who's gone from horses to boys despite her mother's efforts to keep her riding, says, "A lot of boys think horses are stupid, gross and sick."

Tack shop owners, veterinarians and other horse people agree, rather sadly, that they lose business when the girls turn 15 and 16.

Susan Cooper, 15, who lives near Vienna, is a girl they haven't lost. She is slim, with long blond hair, and a choice of boyfriends.She has been riding since she was three years old, and this spring she plans to ride in top-rated pony shows.

This spring, she says, she will cut down on her dating. "Horses take so much out of you. I can't carry a serious relationship with a boy and a horse."

Susan boards her horse for $100 a month at a stable near her house. Her mother, Alla Cooper, says it cost about $225 a month to keep the horse and Susan ready to ride. "It cost a lot more than we ever thought it would," Mrs. Cooper says.

Dr. Davis Hall, a county veterinarian who specializes in treating horses, says that most families under-estimate the cost of owning a horse.

"For many people, a horse is status, especially if the neighbors have a horse," Hall says.

Hall, who came to Northern Virginia five years ago from Texas, says he was surprised at the "ignorance" of many horse owners in the area. He says he often treats horses that are overfed and forced to eat too many food supplements.

Hall says his love for horses is muted occasionally by "spoiled little girls who have no respect for (his) knowledge and time." He's learned not to respond to the little girls who call him in the middle of the night, asking him to chase a horse that jumped a fence.