250 



3. We will also calculate the characteristic functions. If /) represents 

 one of the numbers p„ and q one of the numbers ^„ we have to 

 calculate the following limits: 



K(x;i,).). 



Urn Q* K [x, S, A), Urn \Q*—(-f\ K (x, §, ).), Urn q' —( -^] 



To none of the limits the term =p = \iiithQ{x — S)—sin{){.i- — £,)\ 



contributes. 



For the first of the limits we find immediately 



Urn Q* K (x, §, ;.) 



1 12 



To the second onl^y the term A cosh y (.c — \ I) -\- C cos q {x — ^ I) 

 contributes. First we have 



Urn 



(>' - (^Pli)* 



p-^2/,ll — ^ Q' ^t (q) — ' "osh p cos p 

 and the numerator of the fraction we have found for 



A cosh p (.f — è /) + C cos q (.r — ^ /) 

 changes for q = V// into 



cosh 2 p I — — i 



5 ■ J (eoshp cosp -f- sink p sinp) cosh 2p 



(t-O^'^Kt-OI 



-(- cos 2/>[ - — i ) I cosh 2p j ^ j -|- (cosh p cosp — sinh p sinp) cos2pi~ — { 1 1 



From cosh p siti p -\- sinh p cos p ^0 we have 



cosh p 



cosh p cos p — sin h p sin p = 

 eosh p cosp -j- sinh p sin p 



cos p 



cos p 



cosh p 



and consequently the numerator becomes 

 In this way we find 



lim 



y-{^^^\K(:,,l,X) = 



