"There go our boys down to the south," said a member of the Hurlingham Club in the clipped English of aristocratic London as a stream of propeller-driven cargo planes droned overhead.

The phrase, the accent, and the pride in his voice captured the anomaly that has become a real source of tension this week for the tens of thousands of self-described Anglo-Argentines who find themselves trapped between patriotism and the carefully preserved English breeding brought here generations ago by their ancestors.

While staunchly backing their Argentine homeland in what has become one of its greatest crises, these bilingual, Anglo-Saxon-surnamed thousands here, the outgrowth of long and deep ties between Britain and Argentina, find themselves suddenly suspect and threatened as Britain and Argentina face off over the southern Atlantic Falkland Islands that both nations have claimed for nearly 150 years.

"You have to understand that though I am a patriotic Argentine, the fact that I live in Hurlingham means that many people don't believe that," said another club member who lives in the very British surrounding neighborhood. "We've all been put in an extremely delicate situation."

But for the drone of the aircraft, sharpening the sense of Argentina's military preparations for the conflict that may come, it might have been a normal early fall afternoon at the Hurlingham Club. Couples rested tranquilly on the clipped green lawn, and tennis players swatted serves on grass courts. Inside the ivy-covered brick walls of the Tudor clubhouse, high tea was being served as usual on fine white tablecloths.

Since Argentina seized the islands it calls the Malvinas in an invasion last Friday, the wide network of English clubs, neighborhoods and institutions that have marked Buenos Aires since the early 19th century has been subjected to growing pressure. The Hurlingham Club has been threatened with destruction, and the Buenos Aires Herald, the Anglo-Argentine community's newspaper, has been boycotted by distributors since yesterday for "defending British interests." Actually the paper is owned by Argentines in the majority and a South Carolina newspaper in the minority.

"We have a police guard now," said one worried tea taker in the Hurlingham wood-paneled clubroom today, "but if another dozen Argentines are killed on Monday, those people could help burn the place down."

In the last week, hundreds and possibly thousands of the Anglo-Argentines--estimated to number 100,000 or more--have left for Uruguay, Brazil, or even Britain, fearful of what might happen. For those who remain, the fear of local hostility and Britain's militant anti-Argentine rhetoric have become sources of growing anxiety and resentment.

Britain's strong reaction to the Argentine occupation--and the heated references to Argentine "tyrants" and "dictators" from London--come as a sharp affront to the distinct sensibilities of many of the Anglos here. As Argentines, they say that Britain was unreasonable in the long negotiations over the islands and long ago should have given them up.

And as English-trained Argentines, they are incensed by the idea that the Falklands' 1,800 mostly English residents cannot live happily under an Argentine government.

"There are thousands of Anglo-Argentines living here in Argentina in perfect peace, and they're living very well," said Eric Henderson, an Anglo-Argentine who works in an advertising firm here. "Argentina historically has had many ways of protecting people's rights to live as they please--and the British people here live well indeed." He added, "The British reaction is enormously overstated."

Yesterday, the British community council sent a telegram to British Prime Minister Margaret Thatcher saying that "Argentina has always shown every consideration towards the British community" and that "we do not feel that our situation has been fully considered or taken into account in the difficult problem which has arisen between Argentina and Great Britain."

This kind of sentiment has been welcomed by Argentina's military government, which has pointed to the Anglo-Argentine community as an example of how it can manage the new territory it has seized. Copies of the community council's telegram were widely distributed to reporters yesterday at the Foreign Ministry, which has attempted to assure the Falkland residents that they will retain full rights as Argentine citizens and can retain British citizenship.

While most of the Anglo-Argentine community here have no strong ties to England and many have never been there, English culture has been assiduously preserved in wide areas of the city--from the lawns and hedges of the Hurlingham Club to the private "English" schools, where teachers imported from Britain prepare thousands of Argentine students for both Oxford and the University of Buenos Aires.

"We did the English Cambridge curriculum in the morning and the Spanish-language Argentine curriculum in the afternoon," said a person here who attended an English school. The Argentine curriculum is required by law.

"In geography it was really muddling because we finally ended up thinking when we were young that the Malvinas belonged to Britain in the morning and Argentina in the afternoon."

The history behind this transplanted culture is what may be the source of much of the trouble for the Anglo-Argentines now. In the 19th century, Argentina twice beat off armed British invasions, but nevertheless became a virtual colony of the great European power.

For more than a century, Argentina's key railroad network and much of its large business were controlled by British interests, and its financial system and crucial beef export trade were often manipulated from London. At that time, Buenos Aires was filled with thousands of British citizens, who looked down on the neighboring Argentines and did not allow them to join their exclusive clubs.

Now, a majority of the members of the Hurlingham Club and the patrons of exclusive downtown watering holes like The London Grill and The Richmond Tea Room are Argentines with no English ancestry.

But while English customs and language have become fashionable traits of Argentina's Latin-descended upper classes, many of the Anglo-Argentines say that ever since anti-British feeling began to stir during the 1940s, when Argentina's official position vacillated during World War II, their names have always caused them trouble in times of crisis.

"I'm more Argentine than many people here," said Norman Antelme, another businessman here. "The problem here is that if you don't have a Spanish or Italian surname, you can have problems now. And the awful thing is that we've got nothing else--I'm a loyal Argentine and I can be nothing else."

For other members of the community here, the punishment of the crisis has been more severe. A retired businessman at the Hurlingham Club said he was anxious to leave temporarily but could not because his assets were frozen in a London bank because of Britain's economic measures against Argentina.

"There is all this concern about 1,800 kelpers on the Falklands," said the man, "but here in Argentina we have had our money frozen by Margaret Thatcher and no one cares what happens to us."