76 BELL SYSTEM TECHNICAL JOURNAL 



Similar analysis to points beyond the potential zero yields 



1 , ^'" (^.. 



^^ (2 - Z)i/2 "'"zi/2' ^ ' 



Integration Constant Negative — Types C and D 



Introducing for the constant in equation (7) a negative value, say 

 — {<xV\)^'-, will give a positive value of V (equal to aVi) (or dV/dx = 

 with d^V/dx^ > and must therefore lead to solutions of the C type. 



Integrating once more and introducing the unit So gives 



X = ± 5o(^'/2 ^ 2ai/2)^^i/2 _ „i/2 _|. const. (22) 



Expressing distance from the first plane in units of Sq, we find two 

 possibilities: 



CD = -\- {<p"^ + 2ai/2)V^i/2 _ a'l2 - (1 + 2ai/2)Vl - a"^ (23) 

 and 



(r_ = - (^1/2 _|_ 2ai/2)V.pi/2 _ ^1/2 + (1 + 2«i/2)Vl - ^1/2. (24) 



The first of these solutions gives a potential distribution rising con- 

 tinuously as 0- increases from zero, hence of type D as was anticipated 

 by the subscript. The second solution decreases to a minimum at 



<7min. = (1 + 2al/2)Vl - aW2 (25) 



and then increases, the equation to the right of the minimum being 



(T^ = (^1/2 + 2ai/2)V^i/2 _ c,i/2 + (1 + 2^1/2) Vl - a"\ (26) 



Curves given by equations 23, 24 and 26 are drawn in Fig. 3. If 

 values of a and (p corresponding to conditions on the boundary planes 

 are entered in the figure, a C solution is indicated only if the point 

 falls upon a curve of the cr+ type. This curve then gives the potential 

 distribution to the right of the minimum ; to the left of the minimum 

 the distribution is given by the o-_ curve with the same value of a, 

 which has the interpretation a = (pmin. for this case. Points entered 

 on the o-_ or (xd curves will clearly give D type solutions. 



Three equations for limits of the C region can easily be written 

 down on the basis of the above equations: 



(b) cr = 1 + ^''\ (27) 



(d) 0- = (1 + 2^1/2)^1 _ ^1/2^ (28) 



(e) cr = (^'/2 + 2)V^^/2_ 1- (29) 



