Originally they were awarded for the most important discoveries in the previous year. The time period was changed to five years then shortened to three years. When Victoria came to the throne in 1837 the conditions for the Royal Medals changed to operate on a three year cycle with Mathematics being one of the subjects for which a Medal could be awarded every third year.
In 1850, however, the conditions were changed again as follows:-
... the Royal Medals in each year should be awarded for the two most important contributions to the advancement of Natural Knowledge, published originally in Her Majesty's dominions within a period of not more than ten years and not less than one year of the date of the award, subject, of course, to Her Majesty's approval. ... in the award of the Royal Medals, one should be given in each of the two great divisions of Natural Knowledge.
The list of Royal Medal winners is long, and most, of course, are not mathematicians. The list given below is only a partial list to include those whose biographies are contained in this archive, or are listed among the recent Fellows of Royal Society for whom we have no biography.
MacTutor links:
Royal Society of London
History of the Royal Society
Fellows of the Royal Society
Presidents of the Royal Society
Royal Society Prizes etc:
Royal Medal
Sylvester Medal
Copley Medal
Bakerian Lectures
Other Web site:
The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Societies/RoyalMedal.html