99 



^'V= J// (1 -2e~P' + ^-2/3/) + (- 3 - e-^i^' + 4e- ,^') -f ,9< . (7) 



Consequently for very long periods we find 



— 2k T-i)- kT 



s^ = ^ = —t (8) 



which is the well-known formula for the average distance in the 

 Brownian motion. If we determine the average of (7) witli reference 



,"> 



to all possible initial velocities and if we consider that nj = — . 



2/? 



we find foi' the average square of the distance accomplished as an 



arbitrary initial velocity : 



^«7=-(«^ — l-h.-/30 ...... (7a) 



P 



As long as i^t is lai-ge in relation to 1 — e-^^ the formula of 



Einstein is thus the right one. For the cases, considered in ex[)eri- 



ments, the lowest limit for t to be obtained in this way is of the 



order of 0.01 second. 



§ 2. On the basis of statistical mechanics objections have been 

 raised by Prof. J. D. v. d. Waals Jr. and Miss A. Snkthlagk 'j to 

 the application of the division wiiich has been applied to this case 

 upon the example of Einstein and Hopf in their treatment of another 

 problem. 



Starting from the supposition than in an "ensemble". 



Ku = 

 where K is the active force, they work out another fundamental 

 formula viz. (with a slight variation in notation) 



— = — Q'ti-]-iv ..... . (9) 



where w has to been taken zero. We can again integrate this 



equation and obtain then 



t 

 ü I r 

 u = u^ cos ot -\ ° sin q t -\ \ w (|) sm q (f. — i) d§ . . (10) 







If taking the average we get: 



U =Z U^ COS Q t -f- — sill Q t 



The average velocity would in this way possess a definite period. 

 If however we work out u* we an-ive at an incompatibility. 



1) Gf. Veisl. Kon. Ali. v. Wet. XXIV. 1916. p. 1272. 



