472 EXPLORATION GEOPHYSICS 



If the origin of rectangular coordinates is chosen at the source, 



y2 — ^2 ^ y2 4- ^2 



Also, 

 and 



Furthermore, 



and 



Hence, 



Similarly, 



and 



2r-^—2x or ■^— — 

 d^ ?)x r 





'-""{-r) __ 1 X- 



-dx -dr -dx 



9' 



x^ dr^dx/ dr \?)x^/ 

 d^V^d^'V x\^ -dV/ 1 x^\ 



d!V^d!Vf_^ dV(±_ f\ 

 ^y2 ^^2 j,2 -^rXr r^ J 



9fF ^ 3fF ^ 9F /_!_ _ _£^\ 



9^2 3^2 ^2 + ^^ \^^ ^S J 



yields 



Substitution of these values of -:^^ , ^ y^^^ ^ "^**^ Equation 4 





or 



dr- -dr r 



Since r is the only independent variable in the problem, the partial deriva- 

 tives may be replaced by total derivatives. That is, the last equation may 

 be written in the form 



d'^V _^ 2 dV ^^ 

 dr^ r dr 



This equation can be readily integrated by multiplying by r^, thereby form- 

 ing an exact differential. That is, 



dr^ r dr dr^ dr 



