107 



and in the same way 



As' =i)-T (22) 



In order to arrive (hen at the dilfei-eMtial e(niali(ni for the fmictioii 

 of frequency we reason again in the same way as l)efore. Let 

 /{s,.s\t) represent the chance that a particle that at the time has the 

 coordinate s^ possesses at the time r the coordinate .v (with margin 

 ds) where wo determine the average according to the initial velocity. 

 Now we follow the movement for a short time t and bnild (lie 

 function of fiequention at the time t -\- t tVom that on time s. if 

 Aa' again lepresents the mean deviation dui'ing the time t, and 

 r/(A.s-) the function of tVetiuency, we know for this deviation that 

 we have 



Lp (As) dAs=\, I As (/ (As) dLs = kr and /(As)* v (As) (ILs=i)T (22a) 



We then obtain 



/ (s, s„ t + t) ds = jds'f{s\ s, t) ,1 {Ls)dLs . . . (23) 



where a' =^ s — A.v. 



If now we take (20) into consideration we find for the connection 



of ds' and ds 



/ I dk \ 



ds' = il ^r]ds. 



V 1^ OS j 



Developing according to (23) up to the first order with respect 

 to T, we find 



df 1 d ^ÖY 



ot [3 OS 2 os^ 



If we introduce the value for i) and (i we obtain 

 0/ m d m dy 



i~^--R X^^'f'^+C ^ • . . • (24) 



ot O.T ^a ds bji [la OS 



This equation agrees with that of Smoluchowski, if we take I) 



kT 

 (coefiicient of diffusion of the Brownian motion ). The tactor 



is the If? of Smoi.uchowski i.e. the factor with which the force 



mk must be multiplied in order to calculate the velocity which in 

 a stationary condition was caused by this force. 



By Debyk and his pupil Dr. Tummers ') a differential equation for 



1) Debye, Zur Theoiie der anomalen Dispersion. Verb. DeiUsch I'hys. Ges. X, 

 p. 790. 



J. TuMMEBs, Over eleclrische dubbelbreking, Diss. Utr. 1914. 



