1192 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



In this case no satisfactory interpretation has been found when time 

 variability is present. 



When time variation is absent, we work with the conditions 



(76) 



{H : arbitrary constant) (77) 



3C" (grad hT = (^p^y (grad hf = - ^ (R(ni). 



(78) 



From (78) it is evident that (R 9^ implies K" ^ and grad h ^ 0. So 

 we have 



W + 11 / 



which is of the form 



[grad h{x, y, z)f = 4>{h). (79a) 



Now from (79a) follows 



grad h X grad (grad hY = (80) 



which implies that the vector lines of the field grad h are all straight. Since 

 h is harmonic, this restricts the choice of h to the potential fields associated 

 with a uniform parallel flow, a straight line source, or a point source. Hence, 

 for suitably chosen rectangular coordinates (x, y, z), circular cylindrical co- 

 ordinates (p, dj z) or spherical polar coordinates (r, 6, 0), the only possibili- 

 ties, are respectively 



h= x-^ (grad hf = 1 (81a) 



or 



A = In 1 -^ (grad hf = - = e^ (81b) 



P P^ 



