( 405 ) 



s! 

 —. — ; — - — , combinations yielding tlie same distribution of place. As 



*r ^i- • • • '^'i- 



Bo],TZMANN has siiowii, the denominator may be represented by 



ce by approximation, in which / (,r//j) represents 



the function of distribution of the molecules over the vessel. The 



iiitegral is minimum if /'(,r//c) = 6', so the number of combinations 



is then maximum, or the uniform distribution is the most frequently 



occurring one. To show that the deviation from this distribution is 



not largo as a rule, we may examine how many combinations yield 



. s s s 



a distribution, in which instead of — molecules, 1- x^^ [- a-, etc. 



n n n 



molecules occur in the elements. This number is: 



s! 



-+.<, .'(-+.'0/.. (- 



By putting si z= s^^rh e-^ \/23x etc., we get fur this, taking into 

 accouiit that x^ + .^^ -f • • • '*-■« = '^ '• 



n 



n ^ 



.^+-^"i + i . .- + -^■., + i 



, n.v, \ " / tix.^ 



{[/2jts)n-i 1 + — ' .■.•! + 



Now by approximation 



fi.v, n ,r 



2 „ 2 



nx, nx, ' 

 x,-\ '--] ^ . . . 



'^ 2s ^ 2s 



So the lo(j of the denominator (with the exception of the first 

 factor) is by approximation : 



n^M n n 



:£x H \- - Sx-" — - 2x\ 



2s ^ 2s 2s 



The number of combinations becomes now: 



n 



e -' : 



(^/2jrs)«-i 



if we put 2x^ z= nil'', the chance, that the mean deviation is smaller 

 than u, may be represented by ; 



