470 EXPLORATION GEOPHYSICS 



Also, 



and 



^. 1 (?)V ■dV'\, , 

 P \^x ^x J ■" 



If there are no sources or sinks within the cube, the total current entering 

 the cube must equal that leaving it ; hence, the sum of the currents Ai^, Aig 

 and AJa must equal zero. That is, 



P \?>x -dx J ■" pX-dy -dy J 



p \?)2 ^Z ) ^ 



Division of this equation by ( — j yields 



K'bx Zx )dx \ -dy 'dy ) dy \ ^2 -^z J dz 

 The quantity i — — — ) -t— represents the average rate of change of 



^— in the direction of x. When dx is made to approach zero, this quantity 

 approaches the second partial derivative of V with respect to x. That is, 



lim (-dV 'dV'\ 1 _ 'd'^V 



lim (?>V 'dV'\ 1 ^ 'd'^V 

 dx-^Q\'dx 'dx ) dx dx^ 



Also 



lim (dV ?)V'\ 1 ^ d'V 



dy-^0\dy dy /dy dy^ 



and 



lim (dV dV'\ 1 ^ d^V 



dz^O \dz dz ) dz dz^ 



im (dV dV'\ 1 ^ 

 0\dz dz /dz 



Hence, in the limiting case 



3^F d^V d^V _ 



dx^ 3/-+- ^^: 



2 + ^:^+^ = (3) 



This equation is known as Laplace's equation. It is frequently written in 

 the form 



V'F = (4) 



