232 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



mediate field problem for the mean free time models discussed in Section 

 IIB. This solution is achieved by the following theorem: Given the 

 general equation {IS) for constant mean free time 



"'■ ^ + ^^""^ ^hrH ^(c')/(c')n(x) da,, dc (104) 



and the '^ high field'* equation derived from it by setting the gas temperature 

 equal zero 



dh(c) 



dc. 



m 



y + h{c) ^ ^If ^(C')/i(c')n(x) dQy dC (105) 



and the Moxwellian equation derived from {104) ^V dropping the field 

 term 



(c) = i- ^ ilf (COm(c')n(x) d^y dC (106) 



then the solution /(c) of {104) ^s Ihe convolution of the solution h{c) of 

 (105) and the solution m{c) of {106) : 



/(c) = f h{u)m{c - u) du (107) 



We carry through the proof by constructing explicitly the equation 

 satisfied by the convolution. We replace the running variables c, c', C, 

 C' in (105) by u, u', U, U' and multiply in m(c — u). We get 



aT ^ m{c — u) 4- h{u)m{c — u) = 

 dUz 



^^11 ^(U')^(^')^(c - ^)n(xu) ^fl,' dJj 



We now define f{c) by the relation (107), and integrate the above equation 

 over u. The second member on the left comes out to be /(c). For the 

 first member, we carry out an integration by parts : 



f dh{u) f . , { r.( \ dm{c - u) , 

 / -~-^ m{c - u) rfu = - / h{vL) — ^ du 



J OUz J dUz 



d{m{c - u)) ^^ 

 dCz 



= +fh{n) 



= — / /i(u)m(c — u) du 



__ df{c) 

 dc» 



