relative to Stationary Motions. 271 



transposition of the terms, the equation 



By the insertion of these values equation (16) is changed into 

 gji f' (U - T)tfl =2^ „$ P**-*** + ^sc. (18) 



Ll Jo -it t cl C 



Putting herein, from (12), 

 whence follows 



and 



o . / &' ~ , 



* *, 



we get 



( [lJ' ( U-T)A] = 2p i /81ogf-2^=^-* + sg 8c. (19) 



In this equation, which holds for any time whatever, we will 

 now take the mean values of all the terms. The last sum 

 (which, indeed is independent of the time) is thereby unchanged. 

 The mean value of the penultimate sum is, by hypothesis, for 

 large times, to be put = 0. In the rest of the terms we will 

 only indicate the mean values. We thus obtain 



S,gnU-T)^]=2^Slog*+S^S C . . (20) 



In this we have still to consider more closely the left-hand 

 side. The expression 



is the mean value of U— T during the time from to t } and 

 therefore a function of t which with the increase of t comes ever 

 nearer to the constant value U — T, which represents the mean 

 value for very large times. It does not, however, follow that 

 the variation of this function, denoted by 8 t) must also approxi- 

 mate to a fixed boundary value. We have previously seen that, 

 with a function whose changes consist only of fluctuations of 

 constantly equal magnitude, the variation denoted by S t may 

 take ever greater values with increasing time. Corresponding 

 to this, with a function of the kind now in question, which w T ith 

 increasing time makes ever smaller fluctuations, and thus ap- 

 proaches towards a boundary value, it must be considered possible 

 that the variation denoted by B t makes fluctuations the magni- 

 tude of which does not diminish as the time increases. It would 



