926 



t t 



itv{&) {t — »)dd-z=z( dt j ^^) d^ 



o 



or a8 the zero point of the time is arbitrary, it may be replaced by 

 ito{d) (< f §+ d-) d»') =idti w{d) d». 

 The average value in question may now be represented by 



A B 



— \cos n{t~{-c) — cos nC\ -\ jst'n n(<4-^) — sm 7iS,\ 



n " n 



t T t t-^-t; 



tü{t) jiii») {t— »)dd^=— I dè io{i-\-t) j dt (uiih) d». 



; 



For the sake of simplification the time-unity may be chosen so 

 that the time T is equal to 2:t, thus we find 



jtii») d{^ = 2J 



which once again integrated with respect to / from o to t yields 

 s\ -^ [sin n{t -}-§)- si7t r4\ + 



-] t COS n§. |cosn(f + $) — c^s yi^] t $m n§ . 



11 11* n J 



This expression must subsequently be multiplied by 



w{t -f i) = i:\An sin n {t-\-l)-{- B,, cos n{t + £)| 



n 



and thus integrated with respect to è from zero to 2.t. Then all 

 terms of the product in which n has odd values fall out. At last 

 the average value sought for is given by 



uif)) {t—0) d'9 = -~ ^l --\ "eosnt^ '- t sin nt 



where Cn* = An + Bn^. In the usual way this sum may be converted 

 into an integral, in which the average value — is represented 

 by f{n)^). In the average value described we find in this way 



1) By Planck, Einstein, Laue series of Fourier have been applied in the 

 discussion of questions of probability (e.g. average values). 



