1200 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



It will be noted from (110) that simultaneous absence of recombination 

 and time variation is inconsistent with the defining conditions of this case. 



Case 2: 



^ = 0. 



We shall exclude the possibility o^ tt = because it is included in Section 



I. Then, proceeding as in Case 2 of Section L, we conclude that (P itself is 

 a harmonic function and can be used in place of h. (107) and (109) 

 then become- 



(5l((P)+^^ = (114) 



and 



dV ^ 7(1 - git)] 

 d(9 (P 



or 



V{(9, t) = y[\ - !(/)] In (P + }(/) (}(/): arbitrary function). (116) 



Because — - is harmonic and a function of (P, we have 

 dt 



2(R((P) = -""-Z = E(P - F 

 dt 



(£, F: arbitrary constants) 



whence 



2(R((P) = E(P - F (117) 



(115) 



and 



or 



(9(x, y, z, /) = e-'"m(x, y, z) - t (E ^ C) (118a) 



(P(jc, y, 2, = m(x, y, z) + Ft {E = 0) (118b) 



where m{x, y, z) is an arbitrary harmonic function. 



If recombination is absent, these results specialize to (116) and (118b) 



