40 



G. V. L. Charlier 



It follows directly from the Ä-theoreui tliat a stationary state (= statistical 

 equilibrium) cannot occur before the frequency function has such a form that the 

 equality 



/i',/;'-./i./2 = o 



is satisfied. Suppose /" be the solution of this functional equation. If now, at the 

 time t, the frequency function has the value /, then the ^-theorem states that / 

 must accept a series of values 



that assymtotically approach the value 



This result must subsist for all values of the velocities at the time t. 



Suppose that we at the time i^'^^ have thus obtained the frequency function f^^K 

 The same conclusions now hold good regarding /("^^ as regarding /' above, i. e. 

 whatever values may be assumed for the velocities at the time ß'^^ must assym- 

 totically approach the value f°. On the other hand it is evident that if all bodies 

 at the time t^^'> change the sign of their velocities, so that the body m,. gets the 

 velocity components 



instead of 



before, then the values 



(3) , (3) (3) 



(3) .(3) (3) 



p\ p\ p\ ,/; 



and the corresponding values of H must be described in an inverse order to before 

 i. e. the value of H is increasing instead of decreasing. This remark was first made 

 by LoscHMiDï (1876), 



This criticism, as is the case with most so called paradoxes, is largely a matter 

 of words, though under the discussion lies an important meaning. 



The H-theoreni is a theorem o f statistical mechanics and not o f rational mechanics. 



31. The i7-theorem is based on the computation of the number of passages 

 at given azimuts and distances. This number, however, cannot be exactly computed 

 under the given conditions. The values we have to deal with are mean values. 

 The distribution of coordinates and velocities is supposed to be only approximately 

 known. The real distribution may deviate from this more or less, and it is an 

 essential feature of our reasoning that the elementary values of the coordinates, the 

 velocities, and the time may not be chosen as small as we please, while the coor- 

 dinates etc. are known to us only with the same degree of exactitude as these 

 elementary values. The result is a tnean result and the //-theorem pronounces the 

 exceedingly important thesis that the function H upon an average is decreasing. 



From the definition (the factor 1/N is added) 



H = ^Alog f.fdsdü 



