Statistical mechanics 



41 



it follows that H signifies the mean value of the logarithm of the frequency function, 

 where the mean is taken not only over all values of the velocities but also over 

 all values of the coordinates. The if-theorem states that the average value of log /' 

 decreases with the time, or — though this way of putting it is not fully adequate 

 — that the average value of /' increases. The distribution of the velocities changes 

 from a -fless prohahle^> to a »more prohahle>> distribution. It would be quite possible 

 to advance this thesis as a mechanical principle similar to the principle of the least 

 resistance or other principles in rational mechanics. 



Several authors — Boltzmann^ Maxwell, Gibbs — have, indeed, tried to 

 base the kinetic theory of gases on other grounds than those first used by Clausids 

 and Maxwell himself, and employed in this memoir. Instead of considering a 

 certain given system of bodies they conceive of a very great (infinitely great) num- 

 ber of systems each consisting of the same bodies, but having — from one system 

 to the other — different coordinates and velocities. It is this conception which par 

 préférence is called statistical mechanics (Gibbs). If now the initial coordinates 

 and velocities are varied in all possible manners, it is found that certain kinds of 

 systems (systems of equilibrium) occur much more frequently (infinitely more fre- 

 quently) than others, in like manner as, when drawing balls frona an urn containing 

 an equal number of white and black balls, the proportion '/s (± §) between the 

 number of white and the number of drawn balls is obtained much offener (infinitely 

 more often, if the number of drawings is infinite) than any other proportion. 



If I have not used these considerations as the basis of the present investigation, 

 the reason is, more particularly, that we are thus led into mathematical prolixities 

 and difficulties which are far from being solved. We should indeed have to do 

 with the problem of n bodies in its greatest generality. We should have to do 

 with periodical and asymptotical solutions, with integral invariants and with the 

 most subtle problems of the theory of probability. 



For the present I think the »passage method* is to be preferred. It is mathe- 

 matically simpler and leads comparatively more easily to quantitative results. In 

 order to obtain such results it is, at any rate, necessary to fix the function of force 

 acting between the moving bodies, and there is no doubt that this function of force 

 is in our stellar system nothing but the law of Newton. 



Lnnds Universitets Års.skrift. N. F. Avd. 2. Bd 28. 



