OP ARTS AND SCIENCES. 49 



/(».) = /(?.) s;;[»', .-]+/'&.) 2 *"' [r, rf i]+-^ s,"r ! [<•, '-+ 3 ] 



Similarly, we have 

 /(#„) = /(^ 2 ) 2i? [m + r, m + r~\ + f fa) t* [m + r, m + r + 1] 



+ ^^^ n 7 2 [m + r, m + H-2] + . . . . + ^^[ m+ l 3 » + „] 



== SoT 1 {^r^ SiV [* + r, m + r + j] J- , 



/(# 8 ) =/(gr 8 ) &' [m + n + r, m + ft + r] 



+ /' (fj 3 ) % p 7 l [m + n + r,m + n + r + 1] 



+ (1&L %*-*\m + 7i + r, m + n + r+2] + --- 



+ J 7 ^ \m + n + l,m + n+p\ 



0-l)! L 



= So P 7 1 |^^SiV s [m + « + r, m + n + r + s^, 

 etc., etc. 

 And since /'((/) =f{g), f"(g) =f"(ff), etc., and 



1 = 2 7 ™ [r, r J + Si" [w + r,m+ r] 



+ Si'[m + » + )*, m + ^ + r] H , 



it follows that 

 /(*)=/(«) + * 



+ SoT 1 j^r^Xi"[w + r, m + r + s~\\ 



+ 2 Pji i /"(?») ^ [ m + n + r> m + n + r + s]\+ etc. 



VOL. XXVII. (n. S. XIX.) 4 



