Physics. — " The variability with time o f the distinbutions of Emidsion- 

 particle.s". By Prof. L. S. Ornstein. (Communicated by Prof. 



H. A. LORENTZ). 



(Communicated in the meeting of Maicli 31, 1917). 



Smoi-uchowski discussed this problem in different papers and 

 gave a complete survey of liis work in three lectures ad Göttingen. ^) 



He deduced a formula for the average change of the number of 

 particles in an element, which at the moment zero contains n particles. 

 This formula is: 



Z:„ = (r-n)P, ........ (1) 



where P is the probability that a particle which lies in the element 

 at the time zero, may have come outside in the moment /; whilst 

 I' is the number of particles which at a homogeneous distribution 

 over the whole volume would come to lie in the element in con- 

 sideration. 



Also for the average square with a given number of particles 

 ii at the lime zei'O Smoluchowski gives a formula, viz. 



A'u = [{u — if + n] P' + (n + r) P , . . . . (2) 



from which follows — if the average also is determined according 

 to 11 — 



E' = 2v P. 



These relations are deduced by Smoluchowski with the help of 

 calculations of probability, which "nach Ausfiihrung recht kom- 

 plizierter Summationen (yield) merkwiirdigerweise das einfache 

 Resultat". 



It goes without saying, that it must be possible to attain such 

 a simple result also by a less complicated method. That this is indeed 

 the case I want to demonstrate in this paper. At the same time it 

 will be possible to give some extension to the result. 



1. Let us think the space divided into a great number of 

 equal elements, which we shall mark by the indices i . . x . . k. 

 Let there be at a given moment t = n^ . . n, • • m particles in 



1) Cf. Phys. Zeitschr. 1916, p. 557 and also Phys. Zeilschrift XVI. 1915. p. 323. 



