1255 



in which X will he (he irregular force thai does not depend on the 

 velocity. By multiplication l)y .i; we get: 



m d^x' fd.vy d.r' 



2 dt^ \dt J dt 



We then take the mean values, in which Xx falls out; we 



dx' fdx\ RT 



also put -- =: z and ?/H - = — ^, which yields the differen- 



dt \ dt J N "^ 



tial equation : 



m dz RT 



4 3 rr C </ ^ = , 



2 dt N 



through the integration of which T^angevin arrives at equation (1). 

 Now it is clear that this contains an inconsistency. When equation 

 (3) is multiplied by ./•, and this is then averaged over all the particles, 

 we get : 



w^.tjm- — ö.-rCa .r'. 



The lefthand member is , which (luantitv is therefore smaller 



2 dt ' 



than zero. Hence .r" cannot remain constant, but it must exponenti- 

 ally descend to zero. This means that the Hrownian movement would 

 not always continue, but that the particles would soon be reduced 



^ RT 

 10 lest in consequence of the viscositv. Yet Langevin puts ??/ .r' =: — . 



And it is oidy owing to this inconsistency that he finds the value 

 of {la) for L\ 



Not every derivation of (1(7) which rests on the supposition of a 

 friction against the thermal movement ') of the Brownian particles 

 is open to the above-mentioned objection to the same extent. By the 

 method of Einstein and Hopf, which, originally developed for another 

 problem, was adapted for the calculation of L'' by Mrs. de Haas- 



') 1 shall understand by thermal or true velocity of a particle the velocity 



RT 



that a particle has at a definite moment, fur which therefore holds mx* = --rr • 



A' 



A 



I shall put over against it the yneasiirable velocity. Tliis will be defined by - , 



in which ^^ is the deviation reached in a measurable time t. t will be of the 

 order of magnitude of 1 sec. Of course the mean value of the measurable velocity 

 will be independent of the moment at which the measurement begins. It will, 

 however, be dependent on the length of the interval t, as the mean value of i^ 

 increases only proportional to ^ t. Compare further Remark IV at the end of 

 this paper. 



