S60 Mb MURPHY'S THIRD MEMOIR ON THE 



28. To find the quantity to which V„ is the generating function. 



By Art. 26. 



_ d ^'iRh") ^hd''*^ (Rh") 



I .2.S.,.2ndh;"' 1 .2.3...(2» + l)rfA"^" 

 where R= {l~2h(l — 2t) +h^}~K and h is ultimately equal to 1. 



Forming the equation u==h + &uK we have 



A^ d^f{h)h\ k^ d^f{h).m} 



hence {l-2«(l-2^) + «1-^ ■^^ = ^iZ + ^' ^^ ^ 1T2 <//^' +&c.; 



dh' 



du 

 dh 



^l^^'-^^^^^' rS^F^ = '^' "°"^'^""' °^ ^"" ^" {l-2t.(l-20 + >/'M 

 In like manner, 



«-i{l-2«*(l-2/) + «n-^.^ = 2?A-^+^'|f^ 



/fe^ d\{RM) ¥ d'jRh) 



therefore, 



_^M!!!1(^) = the coefficient of >P«- in ., , ^ f^^ .,, ; 

 1 .2...(2« + l)rf^''+* {1-2m(1-20 + «'H 



(* + 2Am-^)^ 

 consequently, V„ = the coefficient of ¥'^*' in v^ _ ^^(i _ 20 + «'}^ ' 



