Relations between Sir Paul McCartney and Heather Mills have taken a fresh downturn, according to friends.

The news came after a day when the couple were to discuss their daughter's schooling at their divorce hearing.

The former Beatle, 65, is determined that four-year-old Beatrice should not be schooled privately when she starts in the autumn.

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Stalemate: Sir Paul McCartney and estranged wife Heather Mills arrive at the High Court for the fourth day of their divorce settlement hearing today having failed to resolve their row over daughter Bea's schooling arrangements

He wants her education to be like that of his three other children Stella, Mary and James.

Private education, in his estimation, "messes up" children and he hopes Beatrice would gain from being grounded by the experience of state schooling.

Sir Paul has told friends how his other children had "certainly turned out all right".

Stella is a successful fashion designer with stores in such places as London's New Bond Street. Mary is similarly successful as a fashion photographer and James is working in the music industry.

But Miss Mills, 40, is apparently set on a private education.

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Not in agreement: The tense negotiations continue behind closed doors as Sir Paul and Heather Mills try to agree a settlement to end their four-year marriage

The couple have already spent three days behind closed doors trying to thrash out a financial settlement to end their fouryear marriage. They are thought to be heading towards a deal in which Miss Mills will receive £55million.

Yesterday, on day three of the hearings, boxes labelled 'mail' were wheeled into Court 34 at the Royal Courts of Justice by a member of her entourage.

Miss Mills was expected to claim she had received hate mail from the moment she was linked with Sir Paul, 65. She has claimed she became vilified around the world by his jealous fans suspicious of her motives.

A source said: "Heather feels like her reputation has been destroyed by this marriage.

"Paul is seen as a saint and can do no wrong whereas Heather is now a hate figure.

"She does not know what the future holds for her and that is one of her main arguments."

It is already known that one of Miss Mills's main demands is that she and Beatrice have 24-hour protection - paid for by Sir Paul - because she believes they could be at risk from cranks.

Last November she conducted a string of highly-emotional TV and radio interviews in which she spoke about receiving death threats.

In one outburst on GMTV, she confessed she had thought about suicide after receiving death threats. And earlier this week she claimed that her reputation had been destroyed by her marriage to the man regarded as a national treasure.

Yesterday, Miss Mills was being assisted in court by a U.S. lawyer.

She is still representing herself in the hearing after being dropped by the firm Mishcon de Reya for unpaid bills of around £1.5million. Miss Mills is also being helped out by a London solicitor David Rosen.

Her mood yesterday appeared pensive in comparison with the previous day and she did not respond to reporters' questions.

She claims to deserve a payout in excess of £55million because her earning potential has been obliterated.

Before her relationship with Sir Paul she was well-known for campaigning on issues including animal rights and landmines.

The possible £55million deal would be made up of a lump sum of £20million and annual payments of £2.5million each year until Beatrice turns 18.

The couple, who married in 2002, split up in May 2006.

The former Beatle has an estimated fortune of £825million and the divorce settlement could be the biggest in British legal history.

But unless one of them takes issue with the settlement and goes to the Court of Appeal, the figures and details might never be known.

The case - due to last five days - continues today.