( 47 ) 



lie between d\ and .^i -f- (b\ etc. . . . ?r„ and ii\_ -\- (Jii\. i. e. tliat the 

 coiidilion is included in ;i Iwcivi^-diincnsioiial Vdliimc cUMnciit o^' 

 till' ih'.sf (>r<li'i\ llieii ;., n and r aix' Icfl midcliiicd. This \va\ ol' 

 priH'ecdiii.ii' is llial of die kiiielic di»M)r\ of «iases in wliicli we arc 

 tlKM-efore jnslilied in coiisidei-iiiii- llio n(»nual lo die taii,i:<Mil |)laii(' of 

 two ('oliidiii^- iiiolecules lo !»e delerniined \\\ cliaiice. 



if we wish to know diis direction accnrale to the first order. 

 tJKMi tlie 12 coordinales and velocities must W known to the second 

 order. If within this volnnie element we deterndnc the place Itv 

 means of new coordinates 'i' /// c/ .../•,' /r./, fwe niiulil call iheni 

 coordinates of the 2'"' class) which \arv within that element over 

 a Unite i-e,uion, e. U'. fnnn lo 1, then the direction P. ft r is afnnclion 

 of these coordinates of the second class, and they detei-iniii(> the 12 

 coordinates and velocities after impact also to Ihe lirst order. 



p]verv collision brings abont a lowering of the order of determination 

 of the coordinates and the velocities ; everv collision canses a scattering 

 by which the condition of the system becomes (me order less 

 deternuned. In order to know the condition (the coordinates and Ihe 

 velocities) after ii. collisions (at least accnrate to (piantities of Ihe 

 lii-sl order) we mnst know the iiutial valnes of the coordinates and 

 the velocities accnrate to the (y/ -|- ' )'^'' oi'der. The longer the period 

 is for which we want lo ])redict the motion, Ihe higher is the ord(>r 

 which is reqnired for onr knowledge at Ihisinslant. The limit is her(» 

 Ihe |)nre mechainc coJice{)lioji, according lo which liie stale is delermine<l 

 for ever, liecanse Ihe data are deternuned with al>solnle accuracy. 



])OI>tzm.\nn's observation, that a system, whose motion is i-evcrsed 

 really proceeds from a more proitable conditi(ni to a less |)rol»able one, 

 namely to that from which Ihe nalnral system started, and that 

 afterwards conditions are reached, which show again an increasing 

 ])robability, includes the assnm[)lion, that in the iiutial slate the 

 coordijiates and the velocities were determined lo Ihe (2/(i -f~ J )^'' <*''<'^'''' 

 so that the i-exerse motion bi'ings Ihe system after ;/ collisions back 

 lo the iiutial \"olnme element of the lirst order: afterwards the 

 direction of the normal is no longer determined, and the fnrlher 

 process mnst be investigated according lo the i-iiles of the calcidns 

 of |)robabilities. The condition wh(>se validity is recpu red tor the proof 

 of the //-theorem, is not satislied durini;- ihe whole backward course 

 of the process; it is here therefore im[)ossible lo decide anylhing as 

 lo the decrease (n- increase of //. As soon a> the initial stale is auain 



and 



reached Ihe direction of the noiiual ce \ses to he deternuned 



Ihe re(|nii-ed condition is satislicMl. b'rom the fnrtlicr coiu'se we may 



therefore predict with certainty, llial // mnst decrease. 



