449 

 and, tiiiallj, we may show that 



?(3)--|-J 



• log(l-^)»og(l + .^) ^^^ 







"=^ 1/1111 1 \ 



= ^ ^ ^ \T~ Y ^ T ~T"^ * ■ ■ ^ 2n — ij' 



Quite oilier expansions of the quantities ?(2) and ?(3) are obtained 

 by substituting in (1) and (2) 



5 = 1(^/5— 1) = a. 



As we have 



J a-1 

 1 — o = a* and = — a 



a 

 we readily get from (1) 



r/,(a')=:.-log'a f f $(2) 

 and from (2) 



y. («')= - flog' « + im iog« + is(3). 



Now writing — 2 log ri = «, we have 



log (1= y — ^ — -. — , 



M = 1 



and substituting y =^ ^ and also // = « in (3), in order to obtain 



expansions of <r^{<^), Tii'^^) and q^^ia"^), we infer that 



5(-) = *^l„-) + i 





»i=co /a\2n-|-l\ 



UJ ^-^ 2n/ '2/1 + 1 (' 



«= 1 



>f=00 



S(3) = l "-+> 



2 ' -i-' 2n7 ' 2?i-f 21 



H = 1 



The index of the first series representing ?(2) is about yfö. for 

 the other two series it is less than ^. 



Again it is possible to convert the series into delinite integrals. 

 From the power-series in a we find 



