These published finding aids may assist you in your research.
A Guide to the Archives and Manuscripts of the Royal Society by Keith Moore and Mary Sampson (The Royal Society, 1995)
The Royal Society: some account of the Classified Papers in the archives with an index of authors by A H Church (Oxford, 1907)The Royal Society: some account of the Letters and Papers of the period 1741-1806 in the archives with an index of authors by A H Church (Oxford, 1908)
A calendar of the correspondence of Sir John Herschel, edited by Michael J. Crowe, David R. Dyck and James J. Kevin (Cambridge, 1998)
A catalogue of the civil and mechanical engineering designs 1741-1792 of John Smeaton FRS preserved in the Library of the Royal Society by H W Dickinson and A A Gomme, (London, 1950)
The Library and Archives of the Royal Society 1660-1990 by M B Hall (London, 1992) 'Catalogues of the miscellaneous manuscripts and manuscript letters in the possession of the Royal Society by W E Shuckard (London, 1840)
Letters and Papers of Robert Boyle: A Guide to the Manuscripts and Microfilm by M Hunter (UPA, 1992). Microfilm available for purchase from University Publications of America.
List of portraits and busts and catalogue of medals in the possession of the Royal Society (London, 1892)
The Royal Society Catalogue of Portraits compiled by N H Robinson with notes by Eric G Forbes (London 1980)
Access to Archives http://www.nationalarchives.gov.uk/a2a/
National Register of Archives http://www.nationalarchives.gov.uk/nra/default.asp
Wellcome Trust library and archives http://library.wellcome.ac.uk/
Churchill College http://www.chu.cam.ac.uk/archives/
The Newton Project http://www.newtonproject.sussex.ac.uk/prism.php?id=1
Newton manuscripts at the National Library of Israel http://web.nli.org.il/sites/NLI/English/collections/Humanities/Pages/newton.aspx