While we're being pedantic... The Royal Family's official website
goes into much detail about styles, titles and names. I thought this tidbit was particularly interesting:
For the most part, members of the Royal Family who are entitled to the style and dignity of HRH Prince or Princess do not need a surname, but if at any time any of them do need a surname (such as upon marriage), that surname is Mountbatten-Windsor.
Not sure why I've gotten so interested in this. I must be watching too much Downton Abbey