EVERY UNHAPPY FAMILY finds its own way to unhappiness, and so it was with the Loomises. Mr. Loomis was forced to leave his Unitarian pulpit and seek a congregation in another town. His wife, a silversmith whose fierce ambition to leave the world a masterwork rested in her craft rather than in her family, grew every day more frustrated with her husband's pious passivity. Rob, the oldest child, found drugs, and was failing his senior year of high school. Laurel, the 16-year-old elder daughter, found boys; lots of boys. It was almost-- but not quite--enough to explain why Vivienne, perceptive and talented beyond her almost 141/2 years, wrote a letter to her favorite teacher, enclosing a recipe for carrot cake and a description of her life as an episode from the Diary of a Mad Adolescent, and went into her mother's empty studio, looped one end of a length of L.L. Bean rope around her neck and tied the other end to a water pipe at the ceiling, and hanged herself.

Vivienne is first the true-life story of a specific young suicide, and her family, her friends, her notebooks, letters, stories and poems; and then an examination of the rising problem of teen-age suicide in general.

Holly Hickler, one of Vivienne Loomis' creative-writing teachers at the Cambridge School in Weston, Massachusetts, and John E. Mack, a Harvard psychiatrist, ask: Why Vivienne? And why more than 2,000 other teen-age suicides in a given year? With the collaboration of the bewildered Loomises, they piece together the final three years of Vivienne's life in an often moving narrative, informing it with sensitive and frequently insightful analysis.

Mack links Vivienne's suicide to her bouts with depression, an outgrowth of a chronically debilitating and destructive vulnerability of character that few people were aware of, a vulnerability only revealed in her notebooks ("My Private Paper Book") after she died. Her day- to-day behavior obscured her sense of despair during days unrelieved by "moonless nights." Her parents recall how eagerly she completed and sent away an application for admission to a new school; in the loneliness of her solitude she had marked the day with a verse to herself: "And then/ There are times/ When I have nothing/ To look forward to/ In life/ At all/ Like now."

Until recently very little was understood about teen-age depression, and we've learned only a few things. Because of the changes a child goes through during adolescence, a clinical picture of teen-age depression fluctuates wildly when compared to adult depression, and it is difficult to identify the "fixed patterns" of response that are recognizable as abnormal.

The problem of diagnosis among the young is compounded because all adolescents are subject to wide swings of emotion. From this point of view, Erik Erikson's popularized "identity crisis" misleads, too, because it suggests that crisis is a natural adolescent expectation. So how is it possible to detect the clues of depression in an abnormal crisis? What made Vivienne different from her best friend, Anne?

"The thing about Anne," Vivienne wrote in her diary shortly before she died, "is that she understands and feels the same feelings of loneliness, separation, peace and inner riot that I am constantly feeling." Yet Anne is alive and Vivienne is not.

Anyone reading Vivienne's "Private Paper Book" would have recognized problems, yet her parents' stiff moral attitudes and exaggerated respect for a child 's privacy kept them from learning of their daughter's dark and deadly secrets. After her death Vivienne's mother regrets that she had not been a benign snoop: "Had I been the kind of mother who goes up and reads it anyway, I might have gotten very tense and insisted that she have counseling and (I might have) saved her life."

There's subtle indication that the psychiatrist John Mack agrees, though he is too much the disinterested interpreter of human experience to say so. Teen-age suicide, he believes, resides in "an extensive set of determining forces --biological, psychological, interpersonal, familial and social--that build, not necessarily in a regular . . . or orderly fashion, toward the final outcome." One thing leads to another until catastrophe becomes inevitable. He discovers how Vivienne came to take her life, but not why.

The trained Freudian detective, Mack finds the quirks in Vivienne's background that prevented her developing a strong sense of self. She was born on stage, to music: her mother, striving to be dramatic and experimental, invited any hospital attendants who wanted to watch Vivienne's "natural" birth while she lay back to listen to music through a stereo headset. Says Mack: "The delivery room experiment seems to have set the stage for their alienation."

Vivienne's father was self-sacrificing to a fault, but he made the sacrifices for his congregation, not family. Her mother, no less than her father, cherished a magnanimous ignorance of the daughters' sexual experimenting; the parents were so determined to be hip and natural they walked naked around the house. But when the children were overwhelmed by constant "lessons" in sexual morality from family, from peers, from the media, there was no one to turn to for defense and protection. The child perceived her father as the victim of "common crowds and bloody knives," a pathetic figure of a father at the time she craved a man's strength.

Vivienne's sensitive nature prevented her screening the most noxious elements in the world around her. In her speed-up culture she could get no distance from social, sexual and family problems. Death became idealized.

There is, sad to say, no scientific data demonstrating that adolescent suicide can be prevented with a particular treatment. Nevertheless, there may occasionally be something that an interested someone can do. Conclude Hickler and Mack: "The balance is often so delicate that a harsh word or a supportive talk with a friend, teacher or therapist may make, at least for a brief time, the difference between death and life.