350 Ma MURPHY'S THIRD MEMOIR ON THE 



which has been also proved to be the coefficient of A", in the ex- 

 pansion of {l — 2h{l — 2f) + h^\~^, (Section IV. Art. 9), and to be equal 



cl" (tt'Y 

 *^ 1 — oQ — ~Tf^' where t' = \ — t, (Section iv. Art. 2.) 



Then representing by V„ the required function which is reciprocal 

 to f, we have by the preceding article 



where it is obvious that when n' is less than w fiVj"' = 0, and it is 

 only necessary that the coefficients may be so determined, that the 

 same equation may remain true when n is greater than n ; and since 

 one of these coefficients is arbitrary, we may put ^„ = 1. 



Now in general, we have by Section iv. Art. 2. 



x{x~l) {x — 2)...{x-n + \) 



f,Pj'' = {-iy. 



(a; + 1) (a: + 2) (a; + 3)...(;r + w + 1) ' 



hence, i F„ #" +^ = ( - 1 )" { 7 ^^ zr~r ^^-h-r^ — v^ 



■' ^ ^ \(w + a; + 1) (w + a; + 2)...(2« + ar + l) 



_. {n-¥x) {n+x-\)...x . {n+x) {n+x~\)...{x—l) . 1 



~ "*'■ (w+ar+l)(ra+ar+2)...(2w+a;+2)^ '"'"{n+x+\){n+x+9)...{'in+x+S)~ ] 



Therefore, when x is any integer from 1 to x , we must have 



A ^ A X{X — Y) 



2w + X + 2 "■" (2» + a; + 2) (2 w + j; + 3) 



. x{x-\) (;r-2) 



~ "*" (2» + x + 2) (2w + a; + 3) (2w + ar + 4) "^ ' 



and putting for x the successive integrals 1, 2, 3, &c. 

 1 



= l-^„+i. 



2m + 3' 



« , ^ 2 . 2.1 



0=1— .4„+i. X—-—: -r-4„+2 



2« + 4 "+'■ (2w + 4)(2m + 5)' 



^ , . 3 . 3.2 . 3.2.1 



2w+5 ^"""+^- (2w+5) (2w+6) "+" (2«+5) (2«+6) (2m+7)* 



&c. &c. 



