1188 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



I. Solutions with P = P{t), N = 



For N = 0, only Case 2 of the previous section occurs, because the con- 

 dition (P = (implying no carriers!) is of no interest. 



J. Solutions with 3C = 3C(0, N 9^ 

 For grad JC = 0, (30) and (32) yield: 



and 



div grad Ol^ = 2 div ni grad ^ = 0. • (56) 



Taking the div grad of (55) multiplied by *U we obtain 



div grad 1l(R('U) = 

 whence, because of (56) 



— - ^(R(ni) = Fni^ + G (F, G: arbitrary constants) 



eN 



or 



~ (R(U) = FOl + G^"\ (57) 



eA 



Substituting this permitted form for the recombination rate function into 

 (55) we obtain 



2 

 + /7cu^ = -G (58) 



au . ^_ 2 



dt 



whence 



'U = Vf{x, y, 2)c-'^ - G/F {F 7^ 0) (59a) 



or 



ni = \//(:^, y, z) - G/ (F = 0). (59b) 



From (56), /(:r, y, z) is subject to 



divgrad/(jt:, y, 2) =0. (60) 



In summary, if and only if (H(^) has the form (57), there are solutions 

 for which 3C = 3C (t) (arbitrary). The 01 is given by (59) in which/ is an 

 arbitrary harmonic function of x^ y, z. 



