As I said before, there's only been one abdication, to my knowledge, and the Duke of Windsor signed it. As far as I know, that is the constitutional situation. Maybe it would be open to negotiation in the future, after all the right of accession has changed in favour of daughters now.
As someone else pointed out, there has only been one abdication in recent history where the constitution allowed it. It did preclude any descendants (the Duke of Windsor didn't even have any descendants at that point)
The Divine Right of Kings went when King Charles 1st was beheaded, and...
That was the throne of Scotland, not England. James, her son, was the first to combine the two which was why he was James VI of Scotland and James 1 of England.
Here's the copy of the Duke of Windsor's abdication where it says 'I renounce the throne for myself and my descendants'
http://en.wikipedia.org/wiki/File:Edward_abdication.png
As I said, if someone abdicates, they abdicate for their line; ie offspring. Edward VIII/ Duke of Windsor didn't have any children but they wouldn't have become the sovreign if he had.