To examine seriously the proposition that there is an American Establishment requires a solemnity that the late Richard Rovere, The New Yorker's esteemed political writer, might have termed pompous, even if the scrutiny were attempted by the well-informed father-and-son team of Leonard and Mark Silk.

To proceed to prove that proposition, as the Silks set out to do in this scholarly volume, might have alarmed Rovere, whose delightful spoof of the Establishment was published in The New Yorker almost 20 years ago.

As he wrote then in "Notes on the Establishment in America," the Establishment is not a "membership organization . . . that collects dues, issues cards or holds meetings openly . . . . It is a coalition of forces, the leaders of which form the top Directorate, or Executive Committee -- referred to sometimes as 'Central.' . . . The Establishment has always favored foreign aid. It is, in fact, a matter on which Establishment discipline may be invoked."

Leonard Silk, the respected economics columnist for The New York Times , and his son, Mark, a teaching fellow at Harvard, carry Rovere's spoof a non-spoofing step further, 351 pages further, in an informative, occasionally amusing, often lively and sometimes tedious dissertation. They write that the Establishment can "trace its roots to a real religious establishment -- the Unitarian Church in Massachusetts."

The Silks properly treat with wariness onto the quicksands surrounding the Establishment. They view it as "an institution which, maddeningly, seems both to exist and not to exist." The very phrase "Establishment," they say, is "a metaphor without a clear referent." Then, having mapped the slippery terrain, they walk boldly into the quicksand in search of solid footholds.

"We see the Establishment as a collective entity," they write, "a 'third force' in the American polity, the other two being Business and Government . . . Where the aim of business is profit and where the aim of politics is power, the aim of the Esbablishment is disinterest and public morality -- which is why, in our conception, it so resembles a state church."

In other words, the Silks' Establishment encapsulates that elusive force that elusive force that humanistic dreamers have envisaged down through the ages in the philosopher-king syndrome. "The Establishment Thinks Big. It worries not only about the whole nation but about all humanity. It searches after the vast abstractions. It pursues the best" (and, they add, sometimes produces the worst).

That is quite a tall order. It claims for the Establishment disinterested attributed that may raise eyebrows even among the human links of the Establishment chain: the chain that connects Establishment institutions to each other and to the power centers of govenment and business that it both nourishes and feeds off.

Five of these institutions are what one would expect The New York Times , Harvard, the Council on Foreign Relations, the Ford Foundation, and the Brookings Institution. The sixth -- the Council for Economic Development -- has not been heard from much lately. What the authors have to say about it will content any reader with a lingering passion to recall intimate details that led to formation of the CED in 1942.

The CED, claim the authors, was the seedbed for big business' present role in and link to the Establishment. We are treated to four solid pages of quotations from Fletcher Byrom, presently the CED's chairman, analyzing the mood of big business today, dissecting its problems with big government and diagnosing the larger problems of the United States in the world. The question question may arise as to what so distinguishes Byrom. The authors quote him from their September 1979 interview as saying that "[John B.] Connally is the only guy who might be able to lick (Ronald) Reagan if enough of us get on his bandwagon soon . . . . Whether he is the best man or not may not be very important." So much for the influence of the Establishment on the 1980 presidential election.

Far more entertaining than the CED are the sections on the other Establishment institutions chosen for study (somewhat arbitrarily, they concede), and they make excellent reading. One capter, titled "Sulzberger v. Oakes," tells the titillating details of what happened to John Oakes, the former editorial page editor of The New York Times , when he ran afoul of the sentiments of publisher Arthur Ochs Sulzberger.

"Sulzberger v. Oakes" is a hilarious, play-by-play account of frantic long-distance phone calls warning Oakes in Martha's Vineyard that he was about to be aced by Sulzberger on a crucial editorial decision; Oakes' mad rush back to The Times to protect his turf; the publisher's decision -- later revoked -- to allow Oakes to write a signed letter to the editor (himself) attacking the logic of the offending editorial; and a denouncement that ends any doubts about who is in charge of the Establishment at 229 W. 43rd Street.

On its larger theme, The American Establishment leaves questions unanswered because answers do not exist. The concept of an Establishment neatly fitted the English governing class. Its success rested on tradition, aristocracy, institutional control and popular acceptability during the glory days of the British Empire. Perhaps during the first two-thirds of the 20th century the United States could claim to have evolved something vaguely similar, but only the palest copy.

That time has past, leaving behind the elusive term "Establishment," a disembodied concept that the authors have valiantly attempted to endow with bone and muscle.

If they have failed, the failure makes their book no less significant. It simply reduces the size of the canvas on which they display their portraits of influential institutions in contemporary America.