Statistical mechanics 11 



(6) AQAs^" At j As^"d^" db" b" Lo"J\"f^'\ 

 where 



^^..^^ /i" =f(i; ^\ ^i", m;/') 



/2" =/(''; ^^y^^\ «2", «'s"' '^'^'2")- 



In the integral (6) we have zJs^" = du^' äv,^' dw^\ where the velocity com- 

 ponents v^' , u\' accept all possible values, if namely the whole number of 

 stars thrown out from zls/' is to be calculated. 



On similar conditions stars are thrown out from zlsg" and the corresponding 

 number is 



AQAb^" At [ A^;' d'^" dh" h" Lù" f^' f^' . 

 Integrating either of these expression we get 



(7) AÜ At / zJsj" Jsg" rff db" h" ^" f;' f;\ 



which expression hence gives the whole number of stars thrown out in the time 

 Af^. Of these stars a part is thrown in into the dominion zJs^, thus increasing 

 the number of stars in the elementary dominion 



A^ Ab^ . 



To obtain the analytical expression for this number, we have to integrate (7) 

 no longer over all values of u^", v^", w/' and u^", v^", w'\ but only over such values 

 of the velocity components as after the passage give rise to a star situated within Az^. 

 The number of stars thrown in is thus 



=^AÜAtj zls/' zlsj" db" h" Co" Z/'/s". 



As, 



where the integral is to be taken over all values of the variables u^' , w^' and 

 v^', w^' such that the 'velocities after the passage are situated within zls, . 



6. We now obtain for the increase in the number of stars through passages 

 the expression 



(in) — (out) = Ail At / zls," zls/' c?f' db" b" ^" f^' f^' 



(8) 



— AQAt / zls, Aeld'^ dbboy f^ f^ . 



The expression (8) has not yet a form suitable for actual calculations. It may, 

 however,^be3reduced to a much simpler form, as will be shown below. I give here 

 the final expression, which is 



(8*) (in) — (out) = Aii At As, / {f,"f," As, d'^ db b <ü 



where the integration is to, be performed over 



* The number of passages is only half as great at (7), two stars being ^thrown out» at 

 each passage. 



