929 



While calculating' these iiitegTalb we need only take into uccoiint 

 terms, which get (he liighest power of (>—(?„ in the denominator, 

 as only these contribute in a wa} w^orlh mentioning to the result. 

 If we execute the quite elementary calculation we arrive at the 

 result 



. , o - (>„ 



sin 1 



2 



ƒ 



ir(i) sin Q{t-lyii = ^^^ - {An' \^Bu') 



) u {q—QuV 







When Qt is great we can write for this 



00 sin 1 



r 2 tn 



J {Q-QuY 4 







in which A and B are the coefficients of the terms of the series 



2jin 

 for which p„ = (v, consequently = q, or rather the integer that 



lies closest to this. 



As long as A^ -\- B' differs from zero the value of the average 

 in question is proportional to the time. A' -j- B^ is strictly zero, 

 this does not hold good, but there is not a single reason to suppose, 

 that in the Brownian motion the term of whicii the frequency is 

 determined by q should just he missing in the FouRiER-series. But 

 even should it be missing, we should on the basis of the suppositions 

 of VAN DER Waals and Dr. Snethi-agk arrive at the improbable 

 result, that the average value of the velocity of a Brownian particle 

 never reaches the equipartiton value. 



4. In VAN DER Waals' paper it is urged that Langevin's deduction 

 of the formula A' would contain an inner inconsistency. This incon- 

 sistency is held not to appear in the theory that Mrs. Dr. de Haas- 

 fjORENTZ has worked out on the basis of Einstein's formula. And as 

 the starting point according to Einstein and that of Lanoevin are 

 identical, it would be surprising if the one theory would be inwardly 

 inconsistent and the other not, unless Langevin should have made a 

 blunder in calculation. This however is not the case; if we formulate 

 the basis as was done in Ornstein's paper, theie exists no contradic- 

 tion. As well Einstein's theory as that of Langevin rests on the 

 following suppositions 



— = — ti}u-^F (6a) 



dt 



