1198 



^ ^ dnH,{n)TUa ) 

 ü 2" . n! 



where O represents a value between O and 1, we get 



0"HJ.v)Hn{.v) 6'e^-'+-' r r''_~^' f n:i\ f n.i 



n\ J 7i\ ; — * * e 4 uf^v^'cosl xu \cos\ av \ ditdv 





2" . 71 f 







and 



^ ^— ^^ - =zz - - ê^*-<-^' I I e "♦ cos a-w cos «r ^ 



2" . n! ^ J J 



2 , 



dudv 



(2^)/ 

 



— ^r-|-z- lie •• St;/ .('« swi «r ^ dudv 



u 



= — e^^'^'^^ 1 I <" -^ cos xu cosav\e - -\- e J dudt) 



2jt JJ 







°^ ,? «■■'+'•- ƒ Öttc Quo 



-I e""''+''<"^ j I e ■* sin .vu sin av \e ~ — e ^ J dudv. 



2:r J J 







Now% by means of {b), we may write 



6av 'Jar \ . hH" 



I e •^cos.rMVe'* + ^ " yf^'< = "-^-"^^ '* cosdxv 



J 



"» j I x'^A 



e ^sinxuKe"^ — e ^Jdu^:^2vjte •* sin 6xv 





 therefore 



1 n_^}z.^^ 



S = -^é'-'M é 4 ros(«-^.i;)rr/v . . . . (21) 







or 



1 ,,.2_^JL_®^' « 6>^HJx)HJa) 



S— e 1-^^ = V üLjL^J , . . (22) 



\/l—6' 2".w/ 



This result shows that the series is diverging when 6^ = ^. 



7. We shall next determine whether any function whatever of a 

 real variable can be expresse<.l in a series of this form. 



