746 Mr. H. Bateman on some Proble 



ms 



The particular case in which ?' = is well known in the 

 Kinetic Theory of Gases. It was shown in fact by Clausius* 

 that the chance that a single molecule, moving in a swarm 

 of molecules at rest, will traverse a distance x without 

 collision, is 



where I denotes the mean free path or the probable length 

 of the free path which the molecule can describe without a 

 collision. 



The average number of collisions which occur when a 

 molecule describes a path of length x may be taken to be 



X 



equal to j, and so by applying the general formula we find 



that the chance that the molecule experiences n collisions 

 while describing a path of length x is 



hti) 



x/t 



The general formula may also be used in the Kinetic Theory 

 of Gases in quite a different way, as M. von Smoluchowski 

 has shown in an interesting paper published in 1904J. 



Imagine a certain volume in a mass of gas to be geome- 

 trically but not mechanically bounded, and let the number 

 of molecules which w r ould be contained in this volume in a 

 uniform distribution be v. In consequence of the molecular 

 motion the number will sometimes be greater, sometimes less 

 than this mean value. The chance that exactly n molecules 

 are present in the volume at a given time is 



v n e ~v 



The relative momentary deviation h from the mean value v 

 being defined by the equation 



s> n — v 

 o= . 



Smoluchowski determines the mean value of all these 

 momentary positive and negative deviations. Assuming 



* Phil. Mag. [4] xvii. p. 81 (1859). 



t " Uber Unregelmassigkeiten in der Verteilung von Gasniolekiilen 

 nnd deren Einfluss aiif Entropie and Zustandsgleichung." Boltzmann 

 Festschrift, p. 626. 



