928 



— = — ^)'u + ?<' lö = o (given ad u, and uj 



eads to incorrect results. For we get according to their equation 



t»' = u COS Qt -\ Sin Qt ] -\ \ I io{i,) sin Q{t—^) di: \ 



V (^ J Q \ J \ 







and as we shall once again prove further on the last average value 

 is propoitional to t. From the suppositions of van der Waals and 

 Miss Snethlage the remarkable conclusion does reall}' follow, that 

 the velocity of a Brownian particle should increase infiuitelj. 



The proof of the thesis that (5) is proportional to ^ which isonl}' 

 slightly different from a deduction given by Planck already in 

 another connection, runs as follows. The integral may be written 

 in the form : 



t t 



^V{è) ll'(ï/) sw Q (t—i) sw Q{t—-,i) dt dy. 







or if we interchange integrating and determining the average : 

 W{%) W{ti) sin Q{t — $) sin (j(< — i]) d^ di]. 







If now we introduce )j :^ i: -[- if', we get 



IS 



SI 



idi sin o(<— )5 i \V{S,) n^(§ + ip) sin Q(t—^—x^) rfi|, 



-f 



In this form we again introduce for W a F'orRiER-series in which 



W„ = 0, whilst we must take B = 0. 

 We then find for the average value 



W{^) W{^^^,) = 2: {A\,^B\,) cos ~ ni|r 



n J 



So that the integral in question if for the sake of simplification 



2.-Tn 

 we take — ;- := p„ becomes 



I sin Q {t — ^)d^ I 2 {An* + Bu*) sin q (t—i—ip) cos Qu\p di^ - 

 -f 



= i; {Au''-\-Bn^) I sin ^{t — t)d'é | sin Q{t — i — ^) cos Qfi\p dx\> 



