Statistical mechanics 



9 



situated within the dominion Je^ . We say that the star is thrown out from zle^ 

 (using the phrase »throw out» or »in» for passages, and »wander out» or »in» for 

 continuous changes). If As^ varies in all possible ways we get the whole number 

 of stars thrown out from Js^. 



On the other hand some stars, that had not previously velocities belonging to 

 zJej , acquire such velocities through the passages. They are thrown in into the 

 dominion zls^. The difference between these numbers gives the increase within zJs^^. 



4. Number of stars thrown out. All stars have components of velocity 

 equal to u^, v^, and all stars have the components u^, v^, w^. Hence all 

 these stars have the same relative velocity 



^ = V{u2 — ^i)' + (^2 — ^'i? + {''^2 — «(^i)" 

 and the same direction cosines for the relative velocity, namely 



^2 — ^2 — ^1 — ■^i 



We now define a cylinder in the following way: 



Fig. 2. 



1) The axis of the cylinder passes through and has the same direction as 

 the relative velocity and the length =wzl/; 



2) The limiting surface is obtained through turning around the axis a straight 

 linie parallel to co at the distance 



3) The cylinder is limited perpendicularly to the axis by two planes laid 

 through and the other extremity of the axis. 



The dominion limited by this cylinder we describe as the dominion C. 



All stars within G are in the time At passed by at a distance smaller 

 than \ B. The number of passages described by in the time At is hence equal 

 to the number of stars within C. 



Let us compute this number. 



We determine an elementary volume within C in the following manner: 

 We cut out from C two circular cylinders having the axis At and the radii 

 h and h-\-Ah. Further we take two planes including the axis wzl^, which planes 

 with an arbitrary direction, as shown in the figure, have the angles (j; aiid ^l^ + 

 We thus get an element of space having the volume: 



AÜ^=-=iüAtXhA'i^XAh=-ii^AthAhA^. 



Lnnds Universitets Årsskrift. N. F. Avd. 2. Bd 28. 2 



