statistical iiieoiianics 45 



We learn from Meddel. N:o 71 that the convergence of the development (79), 

 supposing only g{u, v, w) to be a positive form of the second degree, is independent 

 of the choice of the generating function f and depends only on the qualities of the 

 function F itself. It is preferable to have a generating function of the most simple 

 form possible and I will put 



{u-L,y-^{i,-L,f-\-iiv-L,y 



where for the present the parameters Z^, L^, are to be left undetermined. 

 Eventually we may put = L.^ = L^ = 0. 

 We have now 



<P = ^1 ?2 ^3 



where 



(83) '^i = -r7H=^ '^^'^ ' 



(X|/ 271 



so that 

 and 



(84) R,,, = Ii}P B^' . ■ ' ■ 

 We have here 



(85) ^ = 



and similar formulae for ^^^d (pg . 

 From (81) we now get 



+ 00 



5,2i+27+2fr rrr 



(86) y-lyfc = I ^. I ^. I ^ / /J Rijk F du dv dw , 



~ GO 



where i?^^, according to (84), may be split into three factors, dependent on u, v, 

 and iv respectively. If F can also be split up in the same way, then the three- 

 fold integral can be exchanged for the product of three simple integrals. 



35. Integration of the fundamental equation. 



The fundamental equation in statistical mechanics has the form 



(87) If =V (/J + □(/!), 



where V(/i) is the »passagef unction » and □ (/i) the »collisionf unction » and f\ is 

 to be considered as a function of t; x, y, u^, v-^, such that 



(88) = M + + -f + Z^. 

 dt dt ' ^ dx ' ^ ^ &Mi dv^ dw^ 



