1260 

 When on acconnt of Zi" = ht , we pnt = 0, and further 



\ — ] := x^ r=z , we find when again we take the mean value 



\dt J niN 



for all particles: 



— TIT 1 



Zi» = ^^ — -t ........ (Ic) 



A ÖJrtJa 



This resnll has half the value of that of equation {\.<i). The good 

 resnlt that Miss Snkthlagk found when calcidating zV from her 

 observations, shows that (la) is to be preferred to {ic). Though 

 accordingly it does not lead to the accurate result, I hope that the 

 above derivation will contribute to give iis a clearer insight in the 

 theory of the Brownian movement. 



When we do not want to stai't from the supposition A* = bt, 



equation (9) can also be solved, when first mean values have been 



taken in a way that strongly reminds of Langevin's. When we put 



/ /// \ * /* 7' 



A''= Ï and ( ) =: we find for ^ the differential relation '): 



\dt J UiiX 



d'i 'e RT 



A — + r* = — ....... (10) 



When we first take this equation without the second member, 

 and when we substitute c,=i'iyt, we get: 



d^ri 1 dn f2r^ 1 \ 



dt' ^ t dt \t 4eJ ' 



When we put besides / = ^ . we find : 



or 



d'ri 1 dv / 1 



.__J J ^ 4- 1 



dr'' X dr V 'T' 



This is Bessel's differential equation for 72 = i, the solution of 

 which runs in the current notation : 



n = AI, (r) ^BY, (t) . 

 Hence : 



L^=\/t[A 1, (2 r \/2t) + B }\ (2 r i/2t)] 

 would be the sohition of the equation without the second member. 



1) The solution of this equation following here I owe to a kind communication 

 from Prof. W. Kapteyn of Utrecht to whom I gladly express my gratitude 

 here. 



