1190 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



This condition clearly satisfies (66) and leads to 



(g(0: arbitrary function). (67) 



%)(«>, = gW + ^ In 

 ae 



(P + ^— 

 a 



The restriction on (P is then provided by the result of substituting (67) 

 into (21): 



divgrad[(P-f ln|5. + ^|] = {(R((P)+i^^]. (68) 



Any (P satisfying (68) constitutes with (67) a solution having the property 

 desired. 



If (65) is substituted into (25) it will be found that the condition 



\ iS /d(P e 



is equivalent to 1 1 = 0, so that Case 1 is characterized by zero total current. 

 Case 2: 



V i8 / acP e 



In this case (66) can be written in the form 



divg«d^^_ a r/ a V_t)_«:i 



(grad (P)2 d(9 W ^ / d(9 e ] ^ ' 



From (69) it follows that (P must be of the form (P {h, t) with 



div grad h{x, y, z, /) = 0. (70) 



In summary we have 



o 



V Theorem 10: liV = 'U((P, /) with grad (P 5^ 0, then either || = or %) = 

 L V{h, t) and (P = (P(A, I) with div grad h{x, y, z, /) = 0. 



We shall investigate the restrictions on the functions /f(x, y, z, /), V{h, /), 

 and (P(/f, in the next two sections. 



Theorem 10 remains unchanged if recombination is absent. If time varia- 

 tion is absent, it simply drops t as a variable in the functions mentioned in 

 the theorem. If both recombination and time variation are absent, the 

 theorem can be strengthened to: 



[Theorem 11: If both recombination and time variation are absent and 

 •0 = V((9), then V = V{h) and (P = (P(A) with div grad h{x, y, z) = 0. 



