( 345 ) 



L 

 Putting na = L, e = — , we have 



00 



Wn {c ; a") = Wn (c/L) ::= J J, (u) -^o f " j du. 



o 

 Now bj raising to the n^^ power the ordinary power series for 



J. (^- J we get 



au 



.>.. _ t=oo(_l)fc /«,A27c Skin) 



where S/c {n) stands for the sum of squares of the coefficients of the 

 expansion [u^ -\- tc^ -{- . . . u,,)^, so that 



S, (n) _ 1 S^ {)i) _ 1 IS, (w) _ 1 3 2 



v.?!-" "■ ^ ' 21 n* ~" 7^ " 2^ ' 3!w« ~ ^ ~" 2^ "^ 3^' 

 Generally supposing 7i very large we may put approximately 



Skjn) ^ 1 



and, substituting, we find that this approximation leads to the suppo- 

 sition 



For small values of ti the approximation is good enough. It is 

 true both functions behave quite differently when u becomes very 

 large, but as they are rather rapidly converging to zero, the actual 

 amount of their difference can be neglected. In particular I find that 

 the integral 



ƒ 



^1 ('') '^Ay) ^^'' 



is of an order of smallness certainly higher than that of the expression 



n-{-\ . . « . , 



2w /2\~2~ /i 



n — 2 \jx J \ct 



while the order of smallness of the integral 



J,{u)e ^" du 



I 



is that of the expression 



1 



1 



