An article yesterday misstated the value of the work performed by volunteers who help the Forest Service. Last year, 43,496 volunteers performed the work of 1,784 persons over the course of the year. For each person-year, the agency received about $13,500 worth of work at a cost of $700.

Wearing slate gray hiking boots, faded jeans and a powder-blue hard hat, Ruth Miller picked up her Forest Service ax and began hacking at a tough old aspen tree.

"Now this," the 50-year-old Richmond, Va., resident said between swings, "has been a wonderful vacation."

Vacation? With a hard hat and an ax?

For Miller and about 50,000 other Americans, summer vacation this year has meant two weeks of unpaid work -- clearing timber and cutting trails -- in the national forests.

These hard-working "vacationers" constitute the Forest Service's burgeoning volunteer corps, the agency's secret weapon in the struggle against ever-tighter budgets for operation and maintenance activities.

Miller's team of 20 workers -- all of whom traveled at their own expense to this cool, dark green wilderness in Southwest Colorado -- was assigned to build a new trail to a popular hiking spot known as "Swampy Pass." The trail was "much-needed," says district ranger Jim Paxon, of Gunnison National Forest. "But we aren't funded to do it ourselves."

In an era of strained domestic budgets, the Forest Service has been one of the leaders in developing a new, cheaper way to get federal work done: volunteerism.

But other agencies, too, have begun to take advantage of a strong desire among many Americans to use their free time to help Uncle Sam do his multifaceted job.

The use of volunteers has been particularly widespread in the "outdoor" agencies, such as the Forest Service, the National Park Service, the Bureau of Land Management, the U.S. Fish and Wildlife Service and the Army Corps of Engineers.

But some desk-bound bureaus, too, have taken advantage of the volunteer spirit. The Veterans Administration has volunteers in its hospitals. The Health and Human Services Department has relied heavily on volunteers in its runaway youth shelters and programs such as Head Start. (Indeed, Head Start has about eight volunteer workers for every paid staffer.)

Even the Internal Revenue Service has turned to volunteer accountants for help when it could no longer afford to provide walk-in assistance for elderly taxpayers.

In addition to volunteer workers, some agencies have made a concerted effort to pull in donations to make up for budget shortfalls.

At the Park Service, for example, Director William Penn Mott has said the next major addition to the national park system -- a preserve of tall-grass prairie land -- will be located wherever the government can find the most land owners willing to contribute their acreage to form a national park.

In essence, the agencies that are relying on volunteers are taking the Reagan administration's drive to turn over more government operations to private contractors one step further.

The administration argues that in many cases, the private sector can provide government services at less cost than federal employes do. But what could cost less than volunteers?

Barbara Passuth of the Forest Service's human resource programs office says her agency gets an average of $13,500 worth of work from each volunteer; the service spends less than $700 -- for equipment, supervision and so forth -- on each volunteer.

A Forest Service study concluded that it would have required $24.1 million in additional appropriations to do the work that 43,496 volunteer forest workers provided in fiscal 1984. This year, the agency expects to have about 50,000 volunteers on the job.

Despite the potential savings, directors of the volunteer efforts at individual agencies say they've gotten virtually no encouragement or assistance from the White House.

The White House has an office to promote volunteer activities -- the Office of Private Sector Initiatives. But a staff member there, who declined to volunteer his name, said no one in the office has any knowledge of agencies' efforts to hold down their budgets by recruiting volunteers.

In the absence of central direction, some agencies have followed the lead of the Forest Service, which has an enabling statute -- the "Volunteers in the National Forests Act" -- allowing it to use volunteer workers and provide injury and liability coverage while they're on the job.

The majority of the Forest Service volunteers are people like Ruth Miller, a physician's wife who decided to donate her vacation to the agency because "I've always loved the forests and I thought it would be a good idea to help."

For two weeks in mid-August, Miller worked a tough eight-hour day, chopping trees, raking back vegetation and tugging at rocks to clear an 18-inch-wide trail through the rugged wilderness. She ate under the clear western stars and slept in a Forest Service tent, 9,500 feet above sea level.

During that two-week period, her team cut three miles of trail up a hillside.

"It would have cost us thousands of dollars to contract out this work," said Shammy Somrak, a Forest Service technician who supervised the job. "Here in the wilderness, where we're not allowed to use any machinery or explosives, getting these trails built is really tough. And here these folks did it for us -- for free!"