There is no complete history of The Royal Photographic Society. The Society’s Minutes of General Meetings 1853- and Minutes of Council 1853-1857 and other archival material to the present day is held on behalf of The Society by the National Media Museum, Bradford.
The Society's Wikipedia entry contains some useful information and the following will also be of help, particularly in relation to the Society's early history:
John Hannavy (editor), Encyclopedia of Nineteenth-Century Photography, London: Routledge, 2008.
J Dudley Johnston, The Story of the RPS [1853-1869], London: The Royal Photographic Society, 1946.
Marian Kamlish, ‘Claudet, Fenton and the Photographic Society’, History of Photography, 26 (4), Winter 2002, pp. 296-306.
Grace Seiberling with Carolyn Bloore, Amateurs, Photography, and the mid-Victorian Imagination, London: Chicago University Press, 1986.
Roger Taylor, All the Mighty World. The Photographs of Roger Fenton, 1852-1860. London: Yale University Press, 2004.
Roger Taylor, Impressed by Light. British Photographs from Paper Negatives, 1840-1860, London: Yale University Press, 2007.
Roger Taylor, ‘Claudet, Fenton and the Photographic Society’, History of Photography, 27 (4), Winter 2003, pp. 386-388