919 



in which P^ is likewise complete; let the domain corresponding to 



this latter be denoted by {^'). We may suppose /3' to be arbitrarily 



little greater than j? — provided y' be chosen su/Jicienthi little greater 



than y — if we assume at the same time, as we did m the previous 



communication, that « and 3 increase and decrease continuoushi w\{\\ 



each other. Let further L^ (y') be the maximum modulus of >]c on 



the circumference of the circle (y'). There is, on account of the 



completeness of Pi and P, mentioned above, corresponding to any 



arbitrarily small chosen number s a whole number E such that for 



ny E 



A.(7')<(^'-7'-f O'S (76) 



together with 



U'"l <(y-« + 0". if |.n<« • • • (77) 



Further we have in the domain («) for all integral not negative 

 values of i and k 



¥!1 < ^!WL .... (78) 



We now suppose m to be chosen greater than 2E and on that 

 supposition divide the double sum (75) into the following four parts, 

 which we denote for brevity by their limits only, 



m — E 00 E — 1 00 "' oo "* ^ — 1 



£"0 m-E+1 E m-E-Irl 



Further we assume e, a/ter y', to be so chosen that y -|- f <C T» 

 say y' = Y -\- f -\- (f. Then we find for the first three sums by means 

 of the inequalities (77) and (78) 



m—E 

 a 



Ö 



«1 |< - ^ w'^• Lt (y') (y — « + e)'«-^- 



E 

 E-1 



,\<~^mkLk (y') (y — « + 0" 



Ö 







m 



h. K ^ f ^--V "^^^ ^n, Lu (y') (V - « + 0'"- *. 



m-E-\-\ 



In the first and third sums moreover the inequality f76) can be 

 applied, and thus we find for s^ 



m—E m 



E ' 



< I 0^— y' + y-« + 2e)'«<-^(^'-.r + 2f)"' 



