317 



J -^ ' (2jr)A+i J e''—l 2A + 2 



o o 



Hence ci„ directly deduced from tlie Benioiillian niiiiiber Bi,^i, 

 is a |)Ositive and rational qnantitj. 



In order to evaluate (lie integral J{z), we write 



00 











and substituting in the remainder ti =: c', we find 



GO CO 



J"" du «— 2mtl/" ^ /^ dv ü«— 2"» 1 



M+« ' g27.l/«_l ^ J v'-(-2 * e27rt._l 27l^TnZ ' 







Hence we have 



00 X 



/du ''='^ , , r 

 — . 2: «~2^fci '« = t 







ZV' 



and, putting n = — , we get 







dv =r 



V 2\/z 







r' 1 



a result from which we deduce at once J (i) = C — |. 



Now according to Stieltjes' theory the integral J {z) can be 

 converted formally in a continued fraction 



11 11 11 11 11 



i—' + r' + i—' + r' + i— ' + .... 



\a^z |a, |a,^ |a^ \a^z 



the quantities Uk depending on the coefficients c„, c^, c,, . . . of the 



21* 



