( 637 ) 



It is then assumed that flie gas is inclosed in a vessel with 



capacity Si, divided into n elements co, so that 7iu>=zil. We imagine 



now a certain distribution of density, at which a^ molecules are 



placed in the first element of volume, a, in the second etc. The 



number of ways, in which N^ molecules may be distributed over the 



n elements, so that every lime tiiis distribution of density exists is 



N! 



— ; . For each of these ways every molecule must be 



a^ / rt, / . . . a„! 



placed in a certain element of volume from the n and so the 



1th 

 representnti\e point" is restricted to the — part of the whole 



generalized space, in the same way with the following, so "the repre- 

 sentiitive points will occupy the fraction 71—^ of the whole of the 

 generalised space". This is in somewhat diffei-ent words nothing but 

 "the chance of each of the combinations is n~'^", and the reasoning 

 rests evidently on the assumption that eacli molecule lins every time 

 an equal chance to any place in the vessel. 



The representative points of the systems with this distribution of 

 velocities occupj^ therefore together a part of the generalized space 



N! 

 = — ; ; .n—"^ (which therefore represents the total chance; an 



rtj 1 a^ ! . . . a„ .' 



expression agreeing perfectly with the ciiance of a certain distribution of 

 denoities in § 1). After a similar reduction as in IjOt.tzmann Ibllows 



from tins : the part ot the generalized space (chance) =r _^— A'A'^^ 



1 ^^ / 1 \ )i(U 



where ^^a^ -r^y^yh -{- ~ \ log — in tlie above mentioned distri- 



s = \ 



bution {A). Now, neglecting { by the side of a, we may consider 

 Ka as a special value of the general function: 



A' = -- I j I — loq — dx dii dz. 



integrated o\er the vessel, where v represents the molecular density 

 as function of the coordinates of an arbiti-ary point, and r the 

 mean density throughout the vessel. A' is a function correspondino- 

 closely with Boltzmann 's N, specially when we leave out the 

 constants and write: 



A' = i \ h' l^!' J' ''•'■ '','/ 'I: just as H was I I Iflofifd.v dy dz ; 

 V is the density function, just as ƒ is the function of ^'elocity. 



