1180 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Finally, by taking the divergence of (7) and making use first of (1) and 

 (2) and then of (5), we obtain: 



o 



[Theorem 9: The vector field 1 1 is solenoidal. 



C. Formulation of Partial Differential Equation System 

 Restricting (P and V 



A very convenient formulation of the partial differential equations re- 

 stricting (P and V is suggested by (18) and (19). Taking the divergence of 

 these equations and substituting (1) and (2) into the results we obtain: 



div grad [nv-^-^(9'^= -a{si + Yf\ (21) 



and 



div ((P grad V) = ^ U + 1 ^ j (22) 



wherein for brevity we have set 



a = f- — 



Mp Mn 



and 



and shall henceforth assume /J ?^ 0, i.e., Hp 9^ fin . Equations (21) and (22) 

 yield immediately a derived equation not containing explicitly the terms 

 introduced by recombination and time variations: 



div grad f iVT) - — (P j = - - div ((P grad V) (23a) 



div (^ + ^ (P) grad V -— grad (P 1 = (23b) 



or 



or 



Either the set (21) and (22) or one of the forms of (23) together with either 

 (21) or (22) constitutes a basic set of two partial differential equations deter- 

 mining (P and V. We are here considering (R as CR((P). 



