112 

 substitution of llie above results in a series of Taylor we find namel}' 



v,=^v 1 



2 AP V* 



Now it is remarkable that, if we follow llie systems back into 

 the time, i.e. determine the average for a moment — rforagrouj) 

 where at (.=:{) the velocity of the suspended particle is v, exactly 

 the same formula can be a|)plied. So that we get a reversible 

 process and questions analogous to the problem of the tops of 

 H curves solved by Ehkknfkst arise. Our reasonings consequently also 

 give in [)rinciple how the objections may be put aside, which Zkkmelo 

 has raised to the statistical mechanics of Gibbs (as well as to the 

 molecular theories of Boltzmann concerning the H theorem '). 



The result obtained may shortly be formulated in this way : the 

 properties of a group of systems, chosen so that in all of them the 

 suspension-particle has a velocity v — a ??-group — , are dependent 

 on the time elajised since the selection. 



We may also ask now after the change of vK with the time for 

 the t'-group selected at the moment ^ = 0. From the preceding- 

 calculation results that 



d „ dv " dK KW v^\ 



dt dt dt M V rV 



from which it follows that the relation 



,7^ = .......... (7) 



which is the right one for the moment in which the group was 

 selected in the ensemble, is not right when this group is followed 

 further. 



It is true that the average for the last member of (6) for the 

 ensemble is equal to zero, as is necessary with regard to ihe 

 stationary character of the whole ensemble, which was already used 

 in the deduction of (3). 



Consequently we shoidd be very careful in interchanging differen- 

 tiation and determination of the average. So equation (5) will only 

 be right for the tirst moment (just as (4)) and consequently also 



^) One ought to bear well in mind that the series-development given liere is 

 only light for a short time after the selection of the r-group from the ensemble. 

 If one follows the group during a long time then the systems of which it consists 

 will have spread themselves over the whole phase-extension with the density that 

 belongs to a canonical ensemble. 



b'or the impoiiance of the Einstein-Langevin formula for this process compare 

 the paper of one of us (Ornstein), (preceding paper). 



