Letters reveal the private Earhart; Letters from Amelia, by Jean L. Backus. Boston: Beacon Press. 253 pp. $14.95.

So much has been written about Amelia Earhart, including three autobiographical reminiscences by ''Lady Lindy'' herself and one book each by her husband and sister, that one wonders what yet another book could add. Surprisingly, this newly published collection of letters from Miss Earhart to her mother has much to offer.

The letters, recently found in four cardboard boxes in the attic of the Berkeley, Calif., house where Amy Otis Earhart spent her last years, show us the whimsical and often touching sides of her famous daughter. They begin with her first written note at age 4 and end with a short letter dashed off to her mother a month before her final around-the-world flight in 1937.

Whether it's a restless Amelia Earhart writing about the restrictions of boarding-school life (. . . ''we're getting a deadly monotone kind of french gray souls and will all be prudes or savages soon . . .''), or an older, wiser public figure advising her mother how to handle the clamoring press (''Reporters may call on you. If so, be pleasant, admit you're my mother if you care to, and simply say you're not discussing plans . . .''), the impression that emerges from these letters is that of a practical young woman who knew what she wanted out of life and knew how best to go about achieving her own goals and those she set for the rest of her family.

That, of course, was one of the recurring criticisms of ''AE,'' as she liked to call herself - the claim that she was domineering and at times patronizing toward both her mother and sister. As the older of the two Earhart daughters, Amelia did assume the brunt of financial responsibility for her family, especially as her father sank deeper into alcoholic despondency and debt over the years. But the care that's apparent in these letters dispels the harsh charges with explanations found in her own words.

While Amelia Earhart's letters home could hardly have avoided the subject of flying and her belief in the safety of aviation and commitment to proving that women pilots were equally as capable as men, the heart of these often brief notes is the resiliency of the relationship between mother and daughter. In her childlike appeals for motherly advice (''What makes sheets smell musty?''), and her even more comical admonitions to her mother regarding clothing and fashion (. . . ''if lying down with knees bent loosen garters to decrease pull over knees. Do not yank any hose on from top . . .''), we get a glimpse of what lay behind one of the few intimate and affectionate bonds that Amelia Earhart ever allowed herself to openly acknowledge.