Tracy Austin, high school senior from Rolling Hills, Calif., gave Andrea Jaeger, high school sophomore from Lincolnshire, Ill., a lesson in back court tenacity last night while winning the $250,000 Colgate Series Championships at Capital Centre.

Jaeger, the best 15-year-old to enliven women's tennis since Austin was that age three years ago, showed her elder no deference, however. She made Austin remember all her lessons and work industriously for the 6-2, 6-2 victory, and the $75,000 top prize.

This was almost entirely a back court battle, rather boring as a spectacle because it lacked stylistic contrast and had some very sloppy patches.

The two talented teen-agers, the youngest players ever to contest a major women's professional final, engaged each other in long baseline rallies, pushing each other from side to side with torrid ground strokes and the odd "moonball" to break up the tempo.

But Austin, at 5-foot-4 and 110 pounds, has a couple of inches and 10 pounds on Jaeger. She is a little stronger, a little more experienced, a little more accustomed to the pressures of a big final. She beat Jaeger for the fifth time in six meetings, and the scores were an accurate reflection of her superiority on this night.

Austin was concerned before the match about a hamstring muscle in her right leg that she pulled in a tournament in November and aggravated during her impressive semifinal victory over Wendy Turnbull Saturday evening.

Her coach, Robert Lansdorp, said before the final that there was a "slight chance, a slim chance" that Austin might not be able to play. But after a few deep knee bends during the introductions and a hesitant first game, she didn't seem hampered at all.

Austin lost her serve in the first game, committing three unforced errors. That was totally uncharacteristic of the tennis she had played in reaching the final with the loss of only nine games in matches against Virginia Ruzici, Hana Mandlikova and Turnbull.

But Austin broke right back in the second game, dug in and was never really in danger the rest of the match. She was broken only once more, when she served for the first set at 5-1.

Austin had more pace and depth and variety on her ground strokes, and was able to sustain her intensity longer than Jaeger, who earned $40,000 in defeat.

The crowd of 10,100 was vocally behind Jaeger, the only player younger than Austin ever to beat her, having done so in the quarterfinals of a tournament in Mahwah, N.J. last August.

But Austin is a tough, composed, self-contained competitor in the Chris Evert Lloyd mold. She kept driving Jaeger deeper and deeper into the corners until she forced an error, or crossed her up by hitting deep and then snapping off a sharply angled shot that Jaeger couldn't reach. The rallies were long and rugged, but almost every time Austin eventually forced Jaeger into a defensive position. Jaeger is a magnificent scrambler, but no matter how hard and deep she hit the ball it kept coming back, and eventually just a maddening inch or two out of reach.

Jaeger got herself into trouble by serving poorly. She only got 64 percent of her first serves in court, compared with Austin's 80 percent. Austin was stepping in and ripping her returns of second serves deep, putting Jaeger under pressure immediately. In fact, Jaeger held serve only twice, in the second and fourth games of the second set.

When Austin broke Jaeger at 30 in the second game, then held serve for 2-1 after four dueces and two break points, her self-assurance grew noticeably. Jaeger lost her serve at love for 1-3, on four errors, and started to get just as noticeably frustrated.

Jaeger lost her serve at love again in the sixth game, and after one unforced error glared at her racket as if she suddenly loathed it. She has a wonderfully expressive way of pouting and chastising herself. She thrusts her arms out, palms up and fingers clenched in self-directed but very public rage, as if leading an invisible chorus of Furies in cursing the dumb shot she has just produced. It is part of her panache.

Austin played a loose game when she served for the set and seemed reluctant to sprint and stretch wide in her backhand corner in an effort to retrieve the crosscourt backhand Jaeger hit at 30-40.That was about the only time Austin's play gave any hint of physical problems, though.

She said afterward she was worried until a late afternoon examination by the tournament physician that she might not be able to play, but once she got on court she felt no limitations.

"I didn't know what to expect," she said. "I didn't know if it would start hurting or get warmed up."

Jaeger missed four of six first serves at 2-5 and was broken at 15 for the set. She seemed to be getting a trifle impatient, rushing her shots, while Austin was taking her time, executing superbly despite her tender hamstring.

The boundlessly energetic Jaeger is a portrait of perpetual motion anyway, her legs and long blond braids blurring together as she dashes around the court, and she never can seem to play fast enough to suit her temperament. But Austin was making her wait, collecting herself before receiving serve, dictating the tempo most of the time between points as well as during them.

Austin asserted her supremacy in the fourth and fifth games of the second set, winning both after lengthy struggles.

She broke Jaeger for 3-1 on her third break point. She made a good "get" to stay in a torrid rally, then Jaeger sailed a forehand long and gave another of her exasperated gestures.

In the next game, Austin held serve after three dueces and won break point. Again the pace of the rallies was blistering, the depth both players got on their shots was admirable, but Austin had a little more variety and strength.

Austin played her best tennis at just the right moments. Whenever Jaeger gave an inkling that she might be getting back into the match, Austin thwarted her, usually outlasting her in a testing rally.

When the points were most important, Austin was right on top of the ball, in position despite her bad leg. She wanted this match and was conceding nothing. She broke Jaeger for the fifth time in the final game, Jaeger sailing another backhand long on the first match point.

"It's kind of weird for me to be the old lady in a big final," said Austin, who turned 18 only last month, in the postmatch presentation ceremonies.

The spectators laughed at that. But they all would agree that on this particular evening she had given the precocious underclassman a solid lesson in how to play the backcourt game.

In the third-place match, Turnbull defeated Martina Navratilova, 1-6, 7-5, 6-2. Turnbull collected $22,000; Navratilova got a $17,000 check.