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THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



terms: in closed form, or as solutions of an ordinary differential equation, 

 or as solutions of a single partial differential equation. The solutions found 

 usually involve arbitrary constants and/or arbitrary functions of various 

 kinds. 



The present section is intended to provide a skimpy but compact sam- 

 pling of the results obtained in the next ten sections. It will be confined to 

 a simple listing of solutions found and furthermore will contain only the 

 forms to which these solutions reduce when recombination and time varia- 

 tion are excluded. (Some solutions are lost under this reduction.) A heading 

 will indicate the section (s) from which the solution comes as well as the 

 formal property associated with each solution. 



For the sake of conciseness and simplicity the symbols denoting arbi- 

 trary constants and functions in this section are independent of those em- 

 ployed in the later sections. They are to be interpreted as follows: 



A , B: arbitrary constants 



h(x, y^ z) : any harmonic function 



(or with subscript) 



{Vf ^): any given solution field 



[G. grad V = 0] 



[H, I. grad (P = 0] 



[J. grad X = 0, .V 9^ 0] 



rv = A 



L(P = h(x, y, 



[V = hix, y, 

 l(P = A 



'V = A + Vh{x,y,z) 



^ Ne /—, r^ 



= ^V/^(x,y,z) 



[K, L. -U = ^{(9), N 9^ 01 



7/ \ I ^ A r^ ~ /i(a;, y, z)"] 

 •U = h{x, y, 2) + ^A -^ — 



{A 9^0) 



