Pete and Jay, they are called, sons of the patrician rich, urbane in the smoothest tradition, politicians in the "new" mold, practitioners of the old art of high expectations.

They invite comparison. "We both worked our way up the hard way - we had to start at the top," Gov. John D. (Jay) Rockefeller IV quips in his office in Charleston, W. Va.

And both he and Delaware Gov. Pierrs S. (Pete) du Pont IX have had all sorts of trouble in their rookie year as governor.

They've been roundly denouced by the press and their fellow pols - including, in Rockefeller's case, meembers of his own party. Legislators have rejected their key programs. And their appointees have gotten them in all sorts of hot water.

"He's almost like Carter, He tried to do too much, too fast," supporters of Rockefeller and du Pont say with remarkable similarity. Or, "Maybe we expected too much."

Part of this can be written off to a series of natural and financial catastrophes Delaware and West Virginia have encountered this year.

But both governors have compounded these difficulties in ways that may come to haunt their thinly-veiled presidential ambitions in years to come.

Du Pont, supposedly an Ivy League smoothie, has stuck his foot firmly in his mouth on at least two embarrassing occasions. His best know slip was to call the average voters of his state "joe Six Packs" - a faux pas that resulted in one Wilmington columnist christening him "Pierre S. Six Pacque IV" thereafter.

He also made a doomsday budget speech to the state legislature during which he declared, "The state of Delaware is bankrupt" - a remark that shook the state's already filmsy credit rating.

In West Virginia, they're still talking about "Jay's Bizzard" - the day Rockefeller went on the radio to warn everyone to head for home because a devastating storm was coming. Thousands heeded his advice, clogging roads all over the state. The storm, however, never came.

Rockefeller acknowledges that he's had a rough year. But then he adds, "This is the second linning of a nineinning ballgame, and what the score is after the second inning doesn't mean anything.

"The most important thing is I'm trying to take on directly the hardest problems this state faces - housing, health, highways - in a way no other governor has been willing to do."

DuPont, 42, who is a promoting himself nationally as a young spokesman for the moderate wing of the Republican Party, is less willing to admit mistakes. "I think it's been a good year," he says. "To tell the truth, I think things have come along a little better than I'd hoped."

If Rockefeller, 40, and du Pont had lived in an earlier time, they'd have been princes of neighboring nation states.

As it is, they're about as close to royalty as anyone in America: holders to two of the nation's best known and wealthiest family names; products of Ivy League educations (du Pont, Princeton and Harvard Law; Rockefeller, Harvard); amtitious young millionaires who've sought their fortune in politics, du Pont as a Republican, Rockefeller as a Democrat in his adopted state of West Virginia.

They even look alike, both tall and slender. So much so that when a news photographer saw Sharon Percy Rockefeller talking with du Pont at a National Governors Conference in Detroit he asked, "Can I get your picture with the governor, Mrs. du Pont?"

They posed smiling, enjoying the mistake. "We'll probably see that picture crop up soon in some magazine," du Pont says. "I don't know if the caption will say Mr. and Mrs. Rockefeller or Mr. and Mrs. du Pont."

At another point when both gobernors were appearing together on the TV news program, "Meet the Press," a reported looked at du Pont and asked him how he was doing handing the problems of West Virginia.

The two millionaires entered office preaching austerity. They cut back on state credit card use and the number of state autos. They halted construction projects. Rockefeller prohibited gilding the State Capitol dome; du Pont called for a reduction in school spending.

Each had won election by a near-record margin, returning their parties to power in their respective states. A lengthy political honeymoon would have been expected for both.

But they soon found themselves in political hot water, embattled in patronage fights and accused of being aloof and inept in dealing with state legislators.

Rockefeller failed in his attempts to deliver on his two top campaign promises: He found little public or legislative support to eliminate the state sales tax on groceries, and his plans to repair pothole-riddled secondary roads had to be cut back when it became necessary for the state to spend millions on flood recovery efforts.

Du Pont's most embarrassing defeat came when his state legislature, unwilling to go along with plans to cut back state spending, overrode his veto of the state budget. It was the first time this had happened in state history.

Rockefeller's setbacks appear more severe than du Pont's. He'd been elected by a landslide, and Democrats held all but 17 of the 135 seats in the state legislature.

"This should have been his most pleasant year, his best of times," says his longtime rival, former Gov. Arch Moore, a Republican. "There's bund to be a period of grace in any new administration. Any new governor brings in a reservoir of goodwill: people want to give you a chance to succeed."

To be sure, many of Rockefeller's early problems were caused by things beyond his control. The winter of '77 brought heavy snow and extremely cold temperatures to the mountain state; roads and airports closed. Energy supplies fell short. Spring brought the worst floods in recent memory, leaving thousands homeless in southwestern West Virginia.

Then too, the new governor got boxed in over patronage. When a new party comes to power in West Virginia, the governor traditionally fires thousands of state workers hired by his predecessor. Rockefeller, cautious over court decisions on the issue and an executive order issued by Moore placing thousands of new workers under the state merit system, was reluctant to do so.

Democratic patronage bosses around the state hit the roof.

"It's a time-honored tradition in West Virginia that to the victor go the spoils," says former Democratic state chairman William Watson. "The party had been out of power for eight long years, and county chairmen naturally assumed Jay could fire all the Republicans and put their men in. They kept waiting for the blanket firings, and they never came."

The outrage among local party bosses has cooled somewhat in recent months, as jobs have slowly opened up. In addition, the governor claims he has reduced state employment by 1,200, toughened mine safety standards, saved millions by eliminating several state construction projects, and that a government reorganization effort he has lauched will pay untold dividends in the long run.

But problems continue to plague him. Black leaders claim Rockefeller has ignored their problems. Legislatiors are uneasy with reports that the governor will ask for a tax increase next year, A Rockefeller plan to build a new West Virginia University football stadium has residents of Morgantown up in arms because they think its location will hopelessly snarl traffic and keep them from getting to a nearby hospital during games. And many insiders have a nagging feeling that Rockefeller has never gotten a grip on his office.

Du Pont's Delaware is far different from West Virginia. It's a tiny state with only three counties and a population about the size of Pittsburgh's, a place long dominated by the du Pont chemical company, which has annual revenues of $4 billion.

Its politics are intensely personal, its legislature a combination of small-town political operatives from the southern part of the state and their big-city colleagues from the Wilmington area.

Du Pont, a former congressman who campaigned under the slogan "leadership for a Change," has had great difficulty adjusting to it, and has spent much of his first year in office warring with his Democratic-dominated legislature. Its members have turned their backs on his budget cuts, refused to confirm many of his appointments ot commissions and boards, and accused du Pont and his aides of being aloof and highhanded.

Then, too, Delaware's financial crisis never materialized quite that way du Pont predicted. The $121 million deficit he forecast in March dropped to $66 million in April, $44 million in May and $40 million in June.

Du Pont blames his problems with the legislators on partisan politics and "the fact that I haven't been willing to cut deals with them." He has stumped the state to build support for his programs. And he called the legislature back into special session twice this fall over two of the state's thorniest issues, overcrowded perisons and court-ordered school integration between the state's largest city, Wilnight, the state's House killed the du mington, and its suburbs. Saturday Pont desegregation bill.

But with a new year approaching, Delaware's financial problems persist and it is likely that a tax increase will become necessary - something that is always awkward when your last name is du Pont or Rockefeller and you are worth millions.

Meanwhile, du Pont is making increasing forays out of the state to talk business and politics. "There's beginning to be some talk here that he might have his eyes on the stars," says one former adviser.