134 Mr. Murphy on the 



serving that the first n - 1 terms may be rejected as not con- 

 taining any negative powers of x, we get 



x" = n x coefficient of - in \ — '- — + ' — =- + &c>, 



x I nx n + l.x* ) 



that is, 



x n = c n + nc n + 1 + -c"- 1 - 2 + t g .c" +3 + &c. 



This principle evidently includes that given in the former section 

 as a particular case, and admits of a proof nearly as simple, viz. 



Put (j> (x) in the same form C.x-a.x-jS.x-ykc. 



and ... L *j£) * i.C" + l.l- a ~ + l.l-; + l.l-?&c. 



x x (3 y 



the part of which containing negative powers of x is 



(a 1 a° 1 a 3 } 



(a; 2 x 2 3 # 3 ^ 



Hence c - is evidently the coefficient of — in — 1. — — , 

 n xx 



1 <$> (x) 



and .. a" = coefficient of - in — nx"- 1 1. -*- — . 



# a; 



Let /(a) be any function of a, as / (0) + A a + Ba + Co* kc 

 putting for n 1, 2, 3 &c. successively, it follows that 



/(„) =/(0) + coefficient of - in -\A + QBx + 3 Cx- kc.\ . 1. ^p 

 = /(0) - coefficient of 1 in /» . 1. ^. 



