1202 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



N. Construction of Solutions from Orthogonal Harmonic Fields, 



N 7^0 



There are many known examples of pairs of harmonic functions hi{x, y, z) 

 and hix, y, z) that have orthogonal vector fields — that is, for which 



grad /fi-grad h2 = (124) 



with grad hiT^O and grad /?2 7^ 0. [E.g., the real and imaginary parts of any 

 analytic function of a complex variable.] From any such pair of functions 

 we can construct the following solutions of (33) and (34) : 



"il = hi; 5C= h2- hi (125) 



and 



(126) ■ 



(127) 



(128) 



The validity of the solution (125) is seen from (33) and this expanded 

 form of (34): 



ni div grad ni + grad ^-grad ('a + 3C) = 0. (129) 



Similarly, the validity of (126) follows from (33) together with a different 

 expansion of (34): 



div grad ^24-2 grad ai-grad 3C = 0. (130) 



It is evident that a given hi and ^2 can be interchanged in the above 

 solutions to yield different solutions, and also that any given hi or ho can be 

 replaced by an arbitrary constant multiple of itself plus a second arbitrary 

 constant. 



O. Construction of Solutions from Orthogonal Harmonic Fields, 



N = 



We can write the differential equation system for the intrinsic semi- 

 conductor [(35) and (36)] in the form: 



div grad (P = (131) 



(P div grad V + grad (P-grad V = 0. (132) 



