103 



the average according to llic. inilial velocily //, and so lind the ensemble 



,ln 

 average ot n — . Then we obtain 

 dt 



du 



u — 



dt 





as u„* possesses the e(|nipartition vabie. Thns it is proved that also 

 for short periods the average vahie of the case average does not 

 come into conflict with statistical meclianics. 



4. 1 shall now dednce the law of frecpiention foi- distribution of 



velocity. If we integrate the eqnation (1) for a short time t, we can 

 write for it 



n — u^z= — i-i u„ T + X or u = u^ (1 — j-fr) \ a; . . (14) 



re x= I F{t) (It and r" = i'^t, 



wheri 







Now there is for ,(■ a law of freqnention '/(.r), so that 



-J- CO ~|~ '^ -f- CO 



I cp (.»;) dx = 1 , \ 'Vff (,v) dx = and I .f' (f (x) dx z= i)T . (15) 



00 — 00 CO 



If now a particle starts with a given velocity ?/„, the nunil>er of 

 particles, for which in the time t the velocity lies l)etween x and 

 u -\- du, may be represented by 



/("o« " 0^^^' 

 or shorter 



f{u, t) du 

 Let us now consider the distribution of velocity at the time/-}-T 

 and again fit our attention on the particles whose velocity lies 

 between u and ii -\- du. These particles have had at t a velocity 

 u' in such a way that 



n' (I — ^t) = u — .1- 



or 



u' — h(1 + iiT) — X ....... (16) 



whilst an interval du' ^ {I -\- t'{T)du corresponds to the interval du). 

 The number of particles that is at t in du' and at ^ + r in du 

 consequently amounts to 



ƒ («', t) (f (x) dx du' 

 and thns we get 



