( 205 ) 







it tbiluvvs, I lull we may write 



.' _i 



r{m) <p (,/■, m) =: ~-^ l{l — ic) 2 ,/„. ƒ'(,,, _ k)j'(-'V<'; m — M '). 



u 



We now cxj)an(l r(///, — i) ƒ f ,r i/',,-, m _ ) and mjike ll.e stih- 



stitutioii 



Then integrating with respect to //', we liiid I he desired expansion 

 of (p {.c, III) in the form 



where jV {j:, m) represents the new series 

 N{.c,m)=:S 



TJie same remarks as were made coneerning tiie lirst integral 

 /'(./', Ill), can here be made again. The integral lias onlv a meaiiing 

 for real valnes of ,v and for posit i\ e valnes of iit, bnt from the 

 expansion is inferred, that the integral ineompleteiy represents a 

 fniiction of .6' which exists over the whole -/.'-plane, qnite ijidepen- 

 dently of the valnes assigned lo the paramelci- m. x\gain the origin 

 ,/; = and ,r ^ X are the only singularilies of the funclion. The 

 singnlarities are logarithmic, when iii is an integer and the origin 

 becomes a regnlar point, when 2 in is cMpiai to an odd integer 



1) It is possible to invert tiii.s reluLion. IL may be sliewu that we have also 

 r (/,/,) /(./•. m) - -^ l/.t r{m - i) = 



14* 



