statistical mechanics 47 



In these formulae we have to assign to i (as well as to j and k) all integer 

 values from 0 to + • If in the first term, in the right member, we exchange 

 i for i' — 1 and in the second term i for i" -\- 1, then i' must accept all values 

 from 4" 1 to -}- and i" all values from 0 to -|- oo , so that 



dx i-J doc 



"pi", /, i- • 



8cc 



We may however even give to i' the value 0 in the first member, if we only 

 suppose A^iji:= 0. Denoting the current indices still by we get 



In like manner we get 



,97.) I = - S - 2 0- + I ) 5^ ... 



and 



(9 - ^ = - L ^''^ -Li^-t 1) Tv. • 



In all these formulae i, j, k go from 0 to + go , and all coefficients Ayk with 

 a negative value of any of the indices i, j or k is put = 0. 

 Further is 



or 



if A^ijk is put equal to 0. 

 In like manner we get 



(98) fJ^^S^^-i.i^-Ty.', 



8r ^''^'^^^ 



(99) ^ ' 



Equating now the coefficients in the left and the right member of (88) we get 

 finally 



dt \ dx dy ds 



(1 00) - f (^ + 1 ) ^iîtUL + (; + !) ?^iiii±l^ + (/^ + 1 ) ^^^'"l 



I call this equation the general recurrence formula in statistical mechanics. 



