of certain Branches of Analysis. 351 



z+ <p z~^ = f^zdz \f^zdz(<p z±(p z~^\+u cl +u c 



^n — ^n J ^ \ n — 2 n — 2 / n— 1 ' 



z= f4>zdz{<p Z ± <p Z~ '\+ Zl C . J"*'' .-— ^- u C 

 ./ V n-2 n-2 ^ n-\ ' 



: f^zdzf^zdzf^ Z + (^ Z )+M C ■ 1 2 • + ^^ ^- . +"n^ 



^ ^ n-3 n-3 "^ n-2 ' n-l 



&C. 



The law of this derivation is immediately obvious, and it may 

 be conveniently expressed in an abbreviatory symbolical form. 

 If * be taken to represent/ \I/ zdz, and w" c be conceived to 

 signify tt^_2<^ we shall have 



(5 Z + t Z~^ = M (C + *) 

 Til — Tn. n \ 



Avhere since f z — fz~^ = 0, u^c and all higher orders of the 

 derivatives of u^ c must be 0. The double sign may be taken 

 from the expression by recollecting that in the case of f z~ 

 or ^o^"^ it is minus. The formula in its simplest and de- 

 finite state is therefore 



We have now only to define the constants u^ c. u^ c, ... u^ c. 

 This it is plain may be accomplished at once by taking z of 

 such a value x, tha't * or f^ zdz =0; for we then have in 

 general <P^x + {-\. f^"^ «„ r" ' = "„ c. (10) 



from which u c may be determined, or at least may be ex- 

 pressed in terms of Uy c. The method of this reduction will 

 lose much of its intricacy and little of its generality, if at this 

 stage of the process we somewhat particularize the functions. 

 We may in conformity with equation (8) suppose 

 ■\)Z dz = — 



and the supposition will give us * an integrable expression, 

 for f^zdz = <P = \ogz. 



Now log z becomes when « = 1 ; we therefore derive 

 from equation 10, 



Mot- = M, c = 2'?, 1 



«jC = «jc = 2^3 1 



U C = Mj c = 2 ^i 1 



u c = u c = 2 (p I 



2m "im+l >/n-fl 



It onlv remains for us to determine 2 <p .1 in terms of 2 ip J . 



2m + 1 



Tluj 



