Undoubtedly, Eton has changed. Boys are no longer admitted because their fathers went there, but because they are clever. And most of them are very clever indeed. The line in the "Eton Boating Song": "At Rugby they may be more clever ..." no longer holds good. This year the school came fourth in the independent school A-level league tables. Parents can put their toddler's name down for a house, but the child will not be admitted unless he is bright. Only the cream of the nation's prep school boys is encouraged to apply and a high score in the Common Entrance Exam is required.
Once they arrive, academic competition is fierce. Arthur Hearnden, secretary of the Independent Schools Joint Council, says: "Eton is not for the faint- hearted. People shouldn't send their children there unless they are sure they can cope with the pressure."
But if their pupils have to be brainier than in the past, most still have to be rich. Fees are pounds 12,300 a year. The school is not part of the Government's assisted places scheme, which helps those from poor backgrounds to attend private schools, but around a fifth receive some help with fees. Mr Hearnden says: "It is much less socially exclusive than it was."
Yet much of the clientele is privileged in one sense or another. A sprinkling of aristocracy continues to arrive each September: the Duke of Montrose's son has just left, and the Duke of Roxburghe's heir started last autumn. Around a third of boys have fathers who were at the school. But if the numbers of the landed gentry have dwindled, those of the nouveaux riches have risen. "It is a hotbed of arrivisme," says one critic. "It is much more about getting into money, making it into the professions, and much less about the old patrician atmosphere of producing a sense of service. Parents who went to minor public schools try to move up a notch by sending their children to Eton."
Class snobbery based on background and wealth may be less prevalent, but it has been replaced by another kind of snobbery based on musical, intellectual or sporting ability. Unfashionably, the school heroes tend to be good sportsmen. An old Etonian who left three years ago claims that the intellectual elite of scholars who come from poor backgrounds but who gained their places through being exceptionally bright and who do not pay fees are not the victims of snobbery. They are just "treated differently. They live in a separate house and they want to move in Goethe and Wittgenstein circles and not the beer-swilling circles of the hoi polloi."
The least acceptable face of modern Eton is a group that flaunts its hedonism and its money. Membership of some school cliques depends on being wealthy. The old Etonian quoted above says: "There are some London-based groups who go to a lot of parties and flash about a lot of money. Equally, there are groups who find this distasteful. The people in my year who had lots of homes were looked down on."
But if the social mix is broader than it once was, Eton remains a bastion of privilege. Its continuing ancient traditions serve to distinguish its pupils from those of other public schools, let alone the state schools that educate 93 per cent of the population. The uniform is hard to take seriously. Etonians have to wear a tailcoat, black waistcoat and pinstripe trousers and a stiff collar. Those who get elected to Pop, the prefects' society, wear grey trousers and any waistcoat they like. House captains wear bow ties. Boys who are late for lessons have their names entered in a "tardy book" and are obliged to report to the school office in uniform early in the morning.
So what are parents paying for? All boys, even the youngest, have their own study bedrooms. The facilities are splendid: beautifully maintained playing fields, two swimming pools, a golf course, a fine theatre, a school of design, and 26 science laboratories. The teaching is very good. Many of the beaks, as they are called, receive more generous salaries than their state school counterparts and some have doctorates. These are powerful attractions.
Mr Hearnden says: "It is the very model of a modern boarding school. It is very liberal and the boys have great independence, and yet the whole atmosphere is orderly and purposeful. It is the nearest thing to a university for boys."
For parents, there are other considerations. Etonians may no longer be assured of top jobs in politics, but both former pupils and parents believe that Etonian connections pave the way for a bright future. A successful stockbroker said recently: "I hated every minute of Eton, but now I'm loving it more and more." As one parent puts it: "If you know people, that is bound to help you get a job, particularly if you want to get into the City. In other professions, it's less useful."
An old Etonian in his second year at university says: "You can't help knowing that you are going to get good connections by going to Eton. I've used it once to jump the queue to get job interviews for holiday jobs. I know it's not correct, and for other jobs I've tried to go legit."
The school that is supposed to have won the battle of Waterloo still plays a powerful role in national mythology. It conjures up images of over-confidence and exclusivity, which sit uneasily alongside the idea of a classless society. It underlines the divisions running through the British school system.
Prince William will have to make his way as a modern monarch. Yet in the school chosen by his parents he will meet only a small selective group. He will, perhaps, learn more about the world than his grandmother, the Queen, who was educated at home, but not much. Eton has changed: it is less aristocratic and more meritocratic, but it is still remarkably different from the schools attended by most of those who will be his subjects.