relative to Stationary Motions. 275 



from which it follows that 



As we have, further, 



$$J = 4>2$i 2 ~ t ~^> 

 the preceding equation changes into 



S, a Z=S,,Z + (J-|)/Z' (29) 



If we apply the kind of transformation hereby exemplified to the 

 last term of equation (28), we get 



(30) 



If now we consider the part of the fraction in (23) which 

 refers to 6, we can give it a simplified form, since according 

 to (26) 



t t 



may be put; and when we introduce into this the preceding 

 expression for 8<p o 0, we obtain 



The first term on the right-hand side of these equations makes 

 with increasing time ever smaller fluctuations and thus approxi- 

 mates to zero ; while the second term makes fluctuations of con- 

 stantly equal magnitude. If, however, we take the mean value 

 of the expression, the second term also vanishes, because the 



difference -^ 1 * s changed into -5 — J. Therefore the part 



of the fraction relative to 6, as well as the part relative to r, 

 fulfils the condition laid down in our theorem, that its mean 

 value vanishes with increasing time. 



This having been established, we can apply to the present case 

 the equation (II.) given in the theorem, and obtain thereby the 

 following— 



S(U-T)=^pSlog* 1 + ^S]ogg+2 <§Sc, . (32) 



U2 



