THE LAUNCHING of the new ballet season in Washington this week is going to be a "family affair" -- the New York City Ballet (which commences a three-week run at the Kennedy Center Opera House Tuesday evening) side by side with its offshoot, the Dance Theatre of Harlem (which arrives at the Warner Theatre Wednesday night for roughly the same period).

The conjunction of two such major visiting ballet troupes is unprecedented for Washington, though it's far from uncommon in the largest dance centers.

The Dance Theatre, for instance, has just returned from a two-week London visit in which it played opposite Martha Graham, the National Ballet of Canada and the London Festival Ballet simultaneously and broke the Sadler's Wells Theatre box-office records notwithstanding.

In this city, however, no one knows for sure how the ballet-going public will respond. When Dance Theatre director Arthur Mitchell learned that the Warner Theatre booking had been inadvertently made without foreknowledge of the City Ballet engagement, he immediately called City Ballet's George Balanchine, whose reaction was, "Why not? By all means."

Mitchell decided to go ahead, though he did scratch a number of Balanchine ballets from the Dance Theatre programs for Washington. "They might prompt unnecessary and misleading comparisons," he says. "We'd rather make sure DTH is taken on its own merits."

The Harlem troupe might well be dubbed Son of New York City Ballet. The company got its start when City Ballet principal Mitchell decided to strike out on his own with the then-revolutionary project of a classical ballet troupe for black dancers. It was an enterprise that had both the approval and assistance of Balanchine, Mitchell's former mentor and the chief ballet master and choreographer of New York City Ballet.

The indebtedness of one company to the other is reciprocal. The memories of City Ballet fans still vibrate in excitement at the thought of Mitchell the dancer. He created major roles in ballets by Balanchine -- "Agon," for example, and Puck in "A Midsummer Night's Dream" -- and by other choreographers. He was among the earliest and most important partners of ballerina Suzanne Farrell. In the company's 40-odd years of existence, he was one of the relatively few dancers whose imprint has been too deep ever to fade.

For his part, Mitchell always has been the first to acknowledge Balanchine as his inspiration. "He is the master," Mitchell says. "He is my teacher. People have said I've imitated him in my own choreography.

Who better to imitate? If I were able to accomplish one-tenth of what he's done, I'd have truly done something."

The Dance Theatre dancers are very much in the long, lean, quick mold of the New York City Ballet, and there are moments even when their dancing appears to be a "purer" realization of the Balanchinean ideal than anyone else's. Despite similarities -- quite a few Balanchine works also are in the repertories of both -- each troupe goes its own, highly distinctive way.

The much larger and older City Ballet is virtually the projection of the wide-ranging artistic vision of three men: choreographers George Balanchine and Jerome Robbins, and the company's intellectual pathfinder, Lincoln Kirstein. The Dance Theatre of Harlem is less a personal reflection of Arthur Mitchell than it is of his philosophy of life and dance. Upon the classical foundations he received at the hands of Balanchine, he has built an eclectic, youthful, contemporary repertoire that runs a choreographic gamut from traditional classicism to the Afro-American heritage to Broadway. The contrasts in idiom and temperament between the two groups should make the twinned visits all the more stimulating.

The Warner engagement will mark the first time the Harlem company has appeared in Washington since its Carter Barron performances in the summer of 1977. Since then, the troupe has undergone a rather radical transformation -- emerging, in Mitchell's eyes, in better shape than ever before.

The Dance Theatre celebrated its 10th birthday this past February, and Mitchell spoke of it a couple of weeks ago when he came down to set things up at the Warner. "It hardly seems possible, and some people don't actually believe it. Because we've established ourselves as a major international company, some folks expect of us the same things they do of a company three or four times our age."

What intensifies the irony is that the troupe was "born again," so to speak, only two years back. Shortly after the Carter Barron visit, fiscal pressures caused the cancellation of a New York season, and that in turn spurred the defection of 11 of the company's (then) 24 dancers, some of whom landed in movie roles or Broadway shows.

"I felt," said Mitchell, "like a parent half of whose children had left home simultaneously, and it certainly was traumatic for the company.

"Most of these youngsters had grown up with me in our school," he said. "They subjected themselves to the discipline all those years, with all the denial that entails, and now they wanted to spread their wings. They'd come to me and say, 'I want to live like a normal person.' What can you do? You can't deny young people. I said, go."

Rumors began to circulate that Dance Theatre was about to fold. "I'm not one to sit down and cry," Mitchell said. "I just took the bull by the horns and made a new company. By now, four of those who left have returned, including Mel Tomlinson, who'd been dancing with Alvin Ailey, and Lydia Abarca, who was on Broadway and in the movie 'The Wiz.'

"On the whole, it's a stronger company now by far. We've learned much more about dealing with young people in the interim, and the new dancers have mostly come out of our own schooling from the start. I think I've grown personally, too, as a teacher and as a director."

Mitchell exudes a palpable sense of pride in looking back on the Dance Theatre's accomplishments in one short decade. "We've done exactly what we set out to do. Initially, we set out to prove to the world that blacks can dance classical ballet. We've done it. If the rest of the world hasn't quite caught up with us yet, or doesn't accept our results, well, that's too bad, but I don't know that there's much we can do about that.

"At the start there was the motivating desire to get the kids off the streets and involved with the discipline of art -- this came from the assassination of Dr. King. Then there was something else: When I made my own breakthrough as an artist and a dnacer, I was constantly being told, you're an exception. No, I said, what's exceptional is that I had the opportunity, and I resolved that I would someday make sure that others had the same kind of opportunity.

"The establishment of the company was the inevitable next step after the school. The young people we had trained needed an outlet, one that didn't exist anywhere else at that time, and they also desperately needed a positive image they could aspire to."

Mitchell wound up with a favorite phrase of his: "You get what you give." In the artistry of the Dance Theatre of Harlem, it can be seen plainly what Mitchell has given -- his all.

The Washington appearances by both dance companies will have unusually enticing aspects. Like Dance Theatre, the New York City Ballet performed most recently in London, where it concluded a higely successful series after some initial, unexpected brickbats from British critics. Mikhail Baryshnikov will be making his final Washington appearances as a member of the City Ballet; by this time next year he will have rejoined American Ballet Theatre as artistic director. In the meantime, he's picked up an Emmy for the public TV program made of his special White House performance during his last Washington visit. He'll be dancing opening night for the City Ballet in Jerome Robbins' new "Opus 19," and later in the week, on Thursday, in Balanchine's "Sonatine," next Sunday afternoon in "Coppelia," and Sunday evening once more in the new Robbins, partnering Patricia McBride in each instance.

The company's other new repertory acquisition to be shown here will be the much touted "Giardina di Scarlatti" by Baryshnikov's friend, dancer Peter Martins (the two of them will just be returntng from Paris performances of the "road show" ballet they mount together). "Giardino," set to 11 of the sparkling harpsichord sonatas of Domenico Scarlatti, calls for 10 dancers in a rococo setting.

Also on deck are the Kennedy Center premieres of Balanchine's "La Somnambula" (Oct. 12 and 13) and "Tchaikovsky Piano Concerto No. 2" (the former "Ballet Imperial," Oct. 11 and 13); an all-Ravel program (Oct. 4), an all-Stravinsky program (Oct. 9), and two evenings devoted to ballets with music by Italian composers (Oct. 12 and 13). Altogether the Kennedy Center repertory will include 24 ballets: 17 by Balanchine, 5 by Robbins, the new Martins, and the Bournonville Divertissements arranged by Stanley Williams. Company principals appearing here, beside those already mentioned, will include Jacques d'Amboise, Merrill Ashley, Suzanne Farrell, Sean Lavery, Kay Mazzo, Francisco Moncion and Helgi Tomasson.

The Dance Theatre of Harlem, which was chosen by Margot Fonteyn -- while the troupe was in London -- to become the only American company appearing in her forthcoming TV series, "The Magic of Dance," also will be offering a plenitude of new ballets. Five works not previously seen in Washington will be shown at the Warner -- Robert North's "Troy Game," Royston Maldoom's "Doina," Carlos Caravajal's "Shapes of Evening," and Mitchel's own "Manifestations" and "The Greatest." A total of 15 ballets will be presented in three separate programs, as well as a fourth which mixes ingredients of the first three.