ELEPHANT MEMORIES Thirteen Years in the Life of an Elephant Family. By Cynthia Moss. Illustrated. 336 pp. New York: William Morrow & Company. $22.95.

For more than 13 years Cynthia Moss followed the histories of 25 or so elephants living in four related families near the center of Amboseli National Park in Kenya. ''Elephant Memories'' is about the park and the researchers who live and work there, but mostly it is the story of Ms. Moss's exotic six-ton neighbors. To write it, Ms. Moss, a former researcher-reporter at Newsweek, became a connoisseur of elephants, learning to recognize individuals by the idiosyncratic geography of their ears, by the tatters, slits and veins. Such traits permitted her to compile a photographic index of Amboseli's roughly 650 elephants. Frequently encountered individuals, particularly the matriarchs who led the families, became as familiar to Ms. Moss as old friends, recognizable at a distance by a shuffling walk.

Avoiding the traditional African-adventure format, Ms. Moss keeps the elephants on center stage, so that she conveys, more realistically than any other popular field account I have read, the day-to-day minutiae of collecting information on who was born, who died, who migrated and who stayed at home, and what they had to eat in the process - all the bits of data needed to reconstruct the demography and ecology of her quarry. Ms. Moss tells the story in a style so conversational, so free of academic jargon, that I felt like a privileged visitor riding beside her in her rickety Land-Rover as she showed me around the park.

One of the project's most interesting scientific discoveries began as a misdiagnosis. When Ms. Moss noticed elephants suffering from a bizarre condition she labeled ''green penis disease,'' she feared an epidemic. With additional research, Ms. Moss and her co-worker Joyce Poole became the first to document for wild African elephants the phenomenon known as must, a ruttish phase in the male reproductive cycle long known to characterize Asian elephants. Each according to his own schedule, bulls undergo annual testosterone-induced Jekyll-to-Hyde transformations from good-natured loners into fiercely aggressive and possessive patriarchs who may actually try to herd the normally independent female groups. Though males attempt to mate throughout the year whenever they happen to encounter a receptive female, a male in must -with streaming temporal glands, a penis turned green and dribbling, strong-smelling urine - does not take no for an answer. The unmistakable smell and long-distance rumblings of a must bull may entice estrous females to approach while warning other males to steer clear - unless they are prepared for a serious fight. Fights between elephants are rare, but between two must males in the same place, inevitable.

Scientific findings about the social behavior of elephants are brought to life in a series of ''semifictionalized'' vignettes. When Torn Ear, matriarch in one of the key families, is killed by poachers, Ms. Moss imagines what must have happened: ''The instant Torn Ear had seen the men, even before the shot rang out, the rest of the herd knew there was extreme danger because Torn Ear had uttered an alarm call just as she charged. All the others heard it and knew who made it and acted accordingly. Most of the elephants immediately began to run away from the source of the danger, but Torn Ear's bond group instantly came to her aid.''

In the ill-fated effort to assist Torn Ear, one of the group's infants, Tina, is shot in the lungs. As the group flees to take refuge near a lake, Tina's female relatives gather round and attempt to buoy her up by working their tusks under her back and under her head. Tina's mother, Teresia, breaks a tusk in the process. Another female ''even went off and collected a trunkful of grass and tried to stuff it into her mouth.'' When the members of the infant's family give up in their efforts to revive the corpse, they still do not leave; they mill about, sprinkling the body with dirt. Some of the females break off vegetation, burying the body with branches and earth.

The elephants stand vigil for most of the night, and only as dawn is approaching do they reluctantly begin to walk away. The mother is the last to leave: ''The others had crossed to the ridge and stopped and rumbled gently. Teresia stood facing them with her back to her daughter. She reached behind her and gently felt the carcass with her hind foot repeatedly. The others rumbled again and very slowly, touching the tip of her trunk to her broken tusk, Teresia moved off to join them.''

Ms. Moss, of course, was not there when poachers ambushed this family. Although I would have liked to know more about the evidence used in reconstructing these events, the fact is that numerous firsthand accounts do exist, telling, for example, how group members held up wounded relatives or attended to dead comrades, events essentially identical to the ones Ms. Moss describes. Almost all the episodes relate known behaviors among these long-lived, highly social and incredibly loyal mammals.

Only on the odd occasion does Ms. Moss permit her empathy for her study subjects to draw her into what is in fact terra totally incognita, as, for instance, when she imagines what the dreams of a dying elephant might be like. Ms. Moss has been tracking Teresia, now matriarch, but who, at the ripe age of 62, having worn out her last set of teeth, is about to succumb from old age: ''She had reached a state of feeling little and she mostly dreamed, perhaps of vast swards of sweet new grass and clear, cool hill streams. Or the taste and feel of the sweet juices that squirted out when she crunched down on her favorite wild fruit. Or most likely the smell and touch and sounds of her family - those who were still alive and others who were long gone, her grandmother, mother, sisters and brothers, sons and daughters.''

''Elephant Memories'' is natural history elaborated so as to invite us highly social human readers to identify with these close-knit family bonds. Ms. Moss has been in Africa since 1968, and writing serves her larger goal - to enlist the reader's support in her passionate desire to preserve elephants. For, ultimately, this is what the book is about: the plight of the elephant in the 20th century. Kenya's population of some 15 million people (according to a 1979 census) will be double that by the year 2000. Inevitably elephants have come into conflict with land-hungry farmers and ranchers who do not welcome such destructive herbivores. Even within the boundaries of the parks, elephants fall victim to poisoning from human debris or fatal ensnarement in garbage pits generated from the immense (and immensely profitable) tourist trade. More gruesome still is the annual slaughter of tens of thousands of elephants shot by bands of heavily armed poachers who supply Japan and other countries with more than 800 tons of ivory each year. And finally, most dreaded of all by Ms. Moss because it will affect some of her closest friends, there is the possibility that park authorities will opt for culling - the systematic legal destruction of entire elephant families in order to save the naturally overgrazed Amboseli woodlands.

When travelers first explored Amboseli at the end of the 19th century, they found a vast, treeless pan with relatively few large animals; indeed, the name ''Amboseli'' derives from the Masai empusel, meaning a barren place. But as this habitat changed in the course of some massive and uncharted ecological succession, growing stands of acacia trees transformed Amboseli into a mixture of savanna and woodland. Elephants and other animals flourished, and Amboseli became renowned as a place to see elephants up close.

But there is no longer room in Kenya for such natural progressions, nowhere else for animals to go while vegetation recuperates. And wild elephants, the ultimate nomads - who, if they stop anywhere but briefly, destroy their own means of survival - are bound to be hard on the habitat. V. S. Pritchett once wrote that if the elephant vanished ''the loss to human laughter, wonder and tenderness would be a calamity.'' Now, barring a rapid reordering of human priorities on a scale more massive than seems possible, that depauperate world is virtually at hand.