statistical meclianics 0 



The velocity component of these stars parallel to the X-axis is u. 

 Let 1) u be positive. Then all stars having an x-coordinate between 



(i) x' — iiJt and x' 



evidently do wander in the dominion AE from without, whereas all stars having 

 x-coordinates between 



(o) x" — uAt and x" 



wander out in the time dt. 



The number of stars between the hmits (i) is according to (1) 



f[x' — ^ uAt) ttAt Ay As Ae 



and the number of stars between the limits (o) 



f{x" — I uAt) uAt Ay As As. . 



But 



f{x' -^uAt)=f{—^ i- ^uAt 



( x + x" \ ( x —x" \ df 



and 



f[x" - i ,At) = / iç^) + if^ - \ uAt^ g . 



The difference between the number of stars wandering in and these wandering 

 out, i. e. the increase in the number of stars, is thus 



— (x" — x') — uAt Ay Az Ab . 

 ^ ' dx 



Writing now Ax instead of x" — x\ we get for the increase the value 



— Atu'^-^AE. 



This result was obtained, supposing u to be positive. Were 2) u negative 

 = — , then we find, for the number of stars wandering in, the expression 



f{x" -j- l u^At) li^At Ay As Ae , 



and for the stars wandering out 



f[x' + \ II ^Ai) u^At Ay Az Ab ' 



and for the difference 



-|- (x" — x') u^At Ay Az^Ab 



= — [x" — a?') uAt Ay Az Az . 



Whether n is positive or negative we thus get for the increase in the number 

 of stars 



— Atu'^-^AE. 



dx 



