Our Queen has, during her long reign, seen off 11 prime ministers.

"Seen off" may be an indelicate phrase, but it is generally accepted that there was at least one that she was quite pleased to see the back of. Guess who? A clue: her longest- serving prime minister. A more obvious clue: her only female one.

David Cameron is the 12th premier to have a weekly meeting with the Queen, generally at Buckingham Palace. What happens on these occasions is of course confidential, though the constitutional expectation is that the premier briefs the monarch on ongoing matters of state, and the monarch offers a little friendly advice in return.

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The Queen's seventh premier, Jim Callaghan, put it neatly: "What one gets from her is friendliness, not friendship".

But there was one premier with whom the Queen apparently got on famously, and that was Mr Callaghan's predecessor, Harold Wilson.

In 1966 the left-wing journalist Matthew Coady revealed that not only did Mr Wilson and the Queen genuinely enjoy each other's company, but that the Queen often extended the weekly audience so that Mr Wilson could enjoy a couple of brandies after the more formal state business had been concluded. Amazingly, Coady's gentle indiscretion prompted something akin to fury. One Oxford history don was moved to describe it as a "constitutional outrage".

It is remarkable that for more than 60 years the relations between the Queen and her head of government have usually been relaxed, and rarely strained. Her first prime minister, Winston Churchill, was genuinely fond of the young Queen, and if he occasionally patronised her a little, she did not let it annoy her.

My personal theory is that the Queen has always wanted to serve longer than her eminent predecessor Queen Victoria. If so, she has just over a year to go now. Victoria, incidentally, was much less discreet than the present monarch, making no secret of her extreme dislike of William Gladstone and her adoration of Benjamin Disraeli. She regarded Gladstone as a demented upstart who was a social disaster. Once, at a posh wedding, Prime Minister Gladstone entered the royal tent. Queen Victoria asked loudly: "Does Mr Gladstone think this is a public tent?"

There is now speculation that our Queen is to begin a constitutional experiment: a kind of dual monarchy, with her heir Prince Charles undertaking more of the monarch's duties and sometimes standing in for her on long-haul state visits. The Queen's media office is to be merged with that of Charles.

This makes sense, as the transition after a long reign is bound to be a tricky process. Yet government ministers will not necessarily be overjoyed at this development, for Charles is already known to be the most proactive member of the royal family when it comes to politics. He is not slow to contact ministers, offering advice and making suggestions. Fair enough, but what happens when the British state enters an exceptionally delicate constitutional period, as it will do if Scotland votes Yes in September?

The White Paper on Scotland's Future states that it is not novel or unique for two independent states to share the same monarch. But the next, exceedingly long, sentence (on page 354) does not make sense, as anyone who reads it carefully will soon realise. I think the implication is that an independent Scotland would be the 17th Commonwealth country to have the British monarch as head of state.

What might worry me would be if the Queen and Prince Charles were "job sharing" during the negotiation period between a Yes vote and the actual arrival of independence. During this sensitive constitutional period, I'm sure the Queen, with all her experience and good sense, would behave with impeccable judgment

I'm not quite so sure about her son. He has already suggested that he would not wish to rule over a divided Britain, though I think he meant divided socially rather than constitutionally.

Meanwhile, the Queen should not formally abdicate. I'm sure she does not want to, and I hope nobody advises her to do so.