Unpublished paper, 'Memoir on the development of functions' by John Walsh Walsh uses three examples with mathematical equations to illustrate the use of the binomial theorem for the development of functions. Covering letter for this paper found at AP/10/15. A previous, abbreviated version of this paper is found at AP/10/14. A graphite inscription at the top right of the first page reads 'Jan /30'. Subject: Mathematics January 1830