46 C. V. L. Charlier 



Considering f\ as a frequency function of type A, we integrate (87) through 

 assuming 



^ ^ ■ ^1 ^ V« ^^^^ 



where 



Ml" + Wj'' 4" W 



gi+J+*{p 



du^^ dv^j dw^ 



k ' 



(90) 'X = " 2a= 



Using the notation 



(91*) nk = 



we have 



(91) = 2" ^y, cpp. . 



The passagefunction as well as the collisionfunction may be developed into 

 similar series, and we write 



(92) . \7if,)^2Vijk9m, 



(93) . \DiA) = 2\Jijk'^ijk. 



Substituting these series in (88) and equating the coefficients of fyk on both 

 sides, we get the required equations for determining Aijk . 



It must be observed that the coefficients Ayi- are functions of the time as well 

 as of the coordinates x, y, z. 



We consider separately the terms in the right member of (88). We get 



(94) . f-V^,^ 



where the right member directly is expressed as an ^.-series. 



Observing that tp^^. only depends on u^^ , iv^ we further obtain 



This development has not directly the form of an ^-series. But 



and according to the recursion formula 



(95) a2i?i+, ^uni\ Œi^x 0 

 (Compare Meddel. N:o 27), we have 



u^Rijk — — a.^RiJ\^ijk — iHi—ijk, 



"l T'jk - '-pi+Kjk ~~ ^ ?i~Ujk 



so that 

 and 



(96) ^ _ N cp,^,,,, -l^.^ fi-.jk 



