1208 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



From (143) and (144) we obtain 



/, = - "^^ 5 (166) 



/. = - ?^» S (167) 



and 



/ = - ^' (m» + l^p) S. (168) 



From (164) and (165) we readily obtain for V^ = 0: 



^ = (I^"J (169) 



and 



e = -*^(p.. (170) 



It is evident that Z = corresponds to the case 3C = constant and implies 



HjJn — fljp = 0. 



The foregoing now provides a formal solution with "Ooo = for every 

 assignment of values to (Poo , Ip , and /„ . There remains the question of the 

 requirements imposed by the condition 



n,p>0 



which is equivalent to 



(P>\N\. (171) 



This implies first of all that (?«, must be chosen > | iV | . 



It is instructive to look first at the case Z = 0. Equation (165) shows 

 immediately that (171) requires the choice of the positive sign for the radical 

 for A^ > and the negative f or iV^ < to avoid (?« < | A^ | . We further find 

 by substitution of (166) and (169) into (165) that (171) requires 



Nip. (172) 



-lnkT^p{(Pl-\N\')_ 



The bracketed factor is positive. Since we are interested only in non-negative 

 values of r, (172) imposes no restriction if Ip is zero or not of the same sign 

 as N. However, for N and Ip of the same sign, (172) establishes an inner 

 radius inside which the solution does not satisfy (171). This may be regarded 

 as establishing the minimum radius for an inner spherical electrode for 



