One has a glowing academic reputation dating back hundreds of years; up to 80 of its pupils a year go to Oxford or Cambridge and only one paper taken by its 210 pupils sitting their GCSEs failed to record an A* to C grade pass this summer.
The other school is an academy that was formerly a state secondary school that twice failed inspections by Ofsted, the education standards watchdog. Last year only just under a third of the pupils got five top-grade passes at GCSE, including maths and English.
Now Eton College has gone into partnership with Langley Academy near Slough. Eton is sharing its superb facilities with its less privileged neighbour – and bringing benefits, including the joys of rowing, to hundreds of state-school pupils for the first time.
As a result of the deal between the two, pupils at Langley, one of the Government's newest academies, will have access not only to Eton's Dornay boating lake and its 450 acres of parkland near the school, but also to the college's 27 cricket pitches.
Langley pupils' sporting skills will therefore be honed on the playing fields where the Battle of Waterloo was reputedly first planned and won. They will also, quite possibly, hear the strains of "Jolly Boating Weather", the opening line of the Eton boating song devised in 1863, as they strive to prove rowing is not just an elite sport for those at independent schools.
Langley, the first state-financed school in the country to offer joint specialisms in cricket and rowing to all its pupils, also has a former Olympic rowing gold medallist, Andy Holmes, on its staff.
The school boasts a museum specialism, too, with its main building decked out with memorabilia – including the canoe used in the Sydney Olympics to win the UK a gold medal.
It's not a bad curriculum to be able to offer to pupils at a school which serves a very mixed catchment area and whose children speak 28 different home languages. Chris Bowler, Langley's principal, reckons that as a result of the deal with Eton, it now offers the broadest range of opportunities of any state-financed school with a comprehensive intake.
Eton's headmaster, Tony Little, is also proud of the partnership between the two schools. "We have personal contacts with the Langley Academy, not least through cricket and rowing," he told The Independent. "We've taken the view that we are happy to be involved with other schools where we have particular interests we can share: cricket and rowing are two of Eton's strengths."
Eton's support for the academy also represents a triumph for ministers. David Miliband, when he was Schools minister, often talked of how he dreamt of the most elite schools in the land becoming part of the Government's programme. "Think about it: the Eton academy, the Winchester academy, it has a certain ring about it," he told a conference at Brighton College in 2002.
Andrew Adonis, the architect of the academies programme as education adviser to Tony Blair, took up the baton when he became Schools minister. He even organised a private dinner with Sir Eric Anderson, former head of Eton and also Mr Blair's teacher at Fettes College, to push the programme.
"I had a very amiable dinner with Sir Eric, who incidentally was at Lincoln College [Oxford] when I was a student there and is now chairman of the school's governing body, and I talked to Tony Little," he said. "We discussed outreach work that the school could get involved with and I put a strong case for an academy."
In a speech to public school heads, pleading with them to back the academies programme – which seeks to replace struggling state secondary schools with privately sponsored schools – Mr Adonis said he wanted to implant the independent sector's DNA into state schooling.
Eton's decision means it has joined a growing line of independent schools supporting academies, including Dulwich College, Wellington College and Winchester. Whilst Eton's involvement stops short of sponsoring the academy (it is being sponsored by millionaire philanthropist Sir Martin Arbib), it has given Langley a great fillip locally.
This year, 520 parents applied to put their children in for the 180 places on offer at Langley, including parents who in previous years said they would have sent their children to the local grammar. Slough is still selective.
Water music: The Eton boating song
Jolly boating weather,
And a hay harvest breeze,
Blade on the feather,
Shade off the trees,
Swing swing together,
With your bodies between your knees,
Swing swing together,
With your bodies between your knees
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