statistical mechanics 



43 



wliere 

 (71) 



which form has the advantage of containing x and a only in the combination x : a 

 (or more precisely {x — L):a). 



33. Special values of the coefficients are given in Meddel. N:o 57. We quote: 



(72) 



S,{x) 



x" 



Sè.^) = —i — 6 + 3 , 



etc. 



Substituting these values in (71), we get the coefficients expressed into the 



moments of the function F{x). 



We get 





To 







Ï1 



Ih" 



a ' 





(73) ^ 



a} 



l^-o" ' 



LË Ta 



\h' 



a 



Ii T4 





6 ^ + 



where all moments are taken about the point L, so that 



(74) 



\[x — LY V{x) dx. 



The moments [i,." about are to be exchanged for the moments [j,,. about the 

 mean [M). The general relation between these moments is: 



(75) i>.," = [X, - r{L-M) ti.,__, + '-^^ {L- M]' \>~,-2 



r{r -l){r-2) 

 1.2.3 



(76) 



Observing tbat \).^ = 0, we thus get 



!J-2" = [^2 + (^— ^I'j'lVo' 



= ÎJ,3-3(L-M)p., -(L-ilf)Vo, 



(x," tj., - 4(L-M) + 6(i:-iJf)V2 + {L -MYi,., 



