The National Gallery of Art, for the first time in its 44-year history, is asking its large audience for small cash contributions to be spent on works of art.

The plea is included in a letter from director J. Carter Brown to be mailed out next week. Although the museum has no membership, the 35,000 people who receive its publications or its monthly calendar soon will hear from Brown.

"The gallery severely lacks adequate funds for purchase," he writes. "Here is a first-time-ever opportunity to express your appreciation."

He adds: "Admission to the National Gallery is free to all. Every event inside it is free . . . Not one penny" received, he assures potential donors, "will ever be used for anything except art acquisition."

The gallery has never received purchase funds from the government. Instead, it has always depended on the generosity of wealthy individuals, particularly founder Andrew W. Mellon and his children, Paul and Ailsa. Both of the gallery's buildings, for example, were paid for by the Mellons.

But Paul Mellon will be 78 in June and, as he puts it, has been "winding down." In 1982, the gallery's trustees began preparing for his departure by soliciting large donations for an art-buying endowment called the "Patrons' Permanent Fund." The principal will be invested, the interest spent on acquisition.

The campaign was designed to raise $50 million by the end of April.

That goal is now in sight: $47.5 million already is on hand.

Brown's letter -- many of whose recipients live in this area -- is designed to attract a sufficient number of smaller contributions to assure the successful completion of the $50 million fund-raising campaign.

The campaign was kicked off with a $5 million contribution from Paul Mellon. Other large gifts -- each in excess of $1 million -- have come from 14 other sources during the last three years. These include the late John Hay Whitney, Walter H. and Leonore Annenberg, Robert H. and Clarice Smith, Ian Woodner, Lila Acheson Wallace, Lois and Georges de Menil, Stavros S. Niarchos, Mrs. Charles W. Engelhard, the Andrew W. Mellon Foundation, the Arnold D. Frese Foundation, Eugene L. and Marie-Louise Garbaty, the Richard King Mellon Foundation, Jill and Arthur M. Sackler, and an anonymous donor who contributed "in honor of Beuford and Teden Cole."

In addition to these seven-figure gifts, the fund has received more than 50 six-figure contributions. Washington corporations have provided more than $1.5 million.

"May 3 is the day on which we shall announce the status of the Fund," he writes, "and any gift that you make that could reach us by April 30 would be particularly welcome, as there is then a good chance that it would be matched in part by a foundation challenge grant."

Brown explained yesterday that funds raised by his letter may be matched in part by the Andrew W. Mellon Foundation of New York.