198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



or 



hp{c) va \dhy_\{c) v — 1 



•(c) 2v — \ \ dc c 



{v + l)a fdK+iic) ,v+2 



K-iic)"! (47) 



where 



V = 0,1,2,3 •'• . 



The auxiliary parameters entering are given by (38) and (39) for equa- 

 tion (46), and (41) and (42) for equation (47). 



The equations (46) or (47) obtained by Legendre decomposition still 

 are, in general, mixed integral-differential equations in one independent 

 variable. Further simplification is possible only in special cases some 

 of which will be discussed later. An even more simple and tractable form 

 of the Boltzmann equation can be achieved in general, however, if one 

 gives up the idea of determining the velocity distribution function and 

 concentrates instead on its moments. In other words, the Boltzmann 

 equation can be looked upon as a system of relations between velocity 

 averages, and as such it becomes a linear algebraic system. 



To carry out this reduction we multiply equation (47) by c'^^ and 

 integrate from to co. The second term on the left is then a simple 

 velocity average. The same is true on the right hand side if two integra- 

 tions by part are permissible and leave no integrated out part, s ^ — 1 is 

 probably adequate for this. The integral over the integral term at first 

 looks as follows 



I f c'^' dc r ^ (^'Y P, (cos k) n(x) sin X dx 



L Jo Jo t{C ) \C / 



In this expression we pass from c to c' as the independent variable. 

 From (41) we see that 



dc _ dc 



7" "7 



Hence the expression becomes 



f c"« ^-gj dc' \ /; {^^ P. (cos .) n(x) sin . .X 



