( 398 ) 



parallel <o cacli otiici- lo and tVo Ix'lwccii Iwu pai-alld walls iioj-iiial 

 k) Ihe (lirectioii of inolioii of llic iiiüleciiles. We assume tlial ])erreclly 

 elastic eollisioii lakes i)lace, and thai the dimensions of the molecules 

 may be neglected. Now if we trace the whole way always in one 

 direction, and introduce the distance to one of the walls as variable 

 /, further consider the velocity to as constantly positive, this case is 

 identical with the i)recedi]ig one, if we call the distance between 

 the w^alls rr, only with this dilierence, that now the distances 

 / 2.T — /, 2jr -|- / etc. come to the same thinj^' as regards the placing 

 of the molecules. So twice as many areas must be taken together 

 as before, and now^ w^e have only to integrate from to Jt. Just as 



before we have now a (luantity s^, = 1 /"A^/ !''(//, which decreases 







because 7^' = 2l I J' 'Io) becomes at last a constant^). 



Such molecules not interfering with each othei- iii their motion, the case 

 of a continuous ensemble of systems of one molecule each, and that of a 

 real gas of n molecules is pretty much the same, just as for the 

 planets. For a gas of three dimensions this is in general no longer 

 the case in conse([uence of the collisions. 



We meet with another ti-ansition case in an ensemble of systems 

 consisting of n planets each. The "entropie tine" is now: 



,^' =: I F In;/ 1' dl ^ . . . (/In dvj^ . . . (/tO,„ 



the Sp is given by 



.".. F'logr' dl,...dln. 







in which I" ^= ^ irdxo^ . . . dv),,, integrated with respect to (o. . . . cy„ 



and summed over all the combinations from I^ to 4 w hich gi\ e the 

 same arrangement of the planets. These condonations are obtained 

 l)y combining the values /,, 2rr -f- /^ etc. with the values /„ 2.t -(- /.^ 

 etc. to I„, '^^ + /ii <^'h'- ii^ '^'1 ways possible. The numbei- of these 

 combinations, so the number of terms in the summation increases 

 indetinitely during the motion, just as in the preceding cases; so 

 /^' becomes constant, and >S^; ai)proaches the minimum \alue. If 



1) Here we might also have taken tlie coordinate .r varying from to tt as 

 variable instead of the coi\linnous /. Then the w sliilts evei y time from + to — 



and vice versa, and we get tlie same lerms j /V/a. but now at the .<ame height .r. 



