ELECTRICAL PROBING WITH AN APPARATUS IN WATER 171 



functions : 



U, = ^f^^ +Ui U, = U[, U, = U^, ..., Un = U'n. (1) 



These functions are partial integrals of the Laplace equation 



To solve the problem we use the Laplace equation in a cyhndrical system 

 of coordinates. The origin of this system we put at the point A, the axis z 

 is put vertically downwards. 



The problem has axial symmetry with respect to the z axis, the Laplace 

 differential equation in this case, does not therefore depend on the airgle 99, 

 placed in a horizontal plane, and has the following form: 



'^2 



-U 1 9U 9^U ^ 



The potential functions Uq, U^, U2, ..., U^, apart from the fact that 

 they should be integrals of the Laplace equation, should satisfy the follow- 

 ing conditions. 



1. The fvnictions Uq\ U^, C/g, U^, ..., U^ should be finite at the points 

 situated at a finite distance from the source of the current and change to 

 zero for points an infinite distance away, 



2. Wheni?->0 



^» 47r R' 



3. For each boundary of separation there should not be jumps in the poten- 

 tial and the normal components of current density should be equal, i.e. 



Ui = f/j+i and ^- = ^- . 



Qi 3z Qi+i 3z 



4. At the boundary of separation of water — air, the normal component 

 of the current density should be zero, i.e. 



1 9U, 



Qo ^z .= 



= 0. 



The solution of the Laplace differential equation is obtained by the known 

 method of dividing variables. A general integral of the equation is the ex- 

 pression 



CO 



f/ = / [5e"^^ /o {mr) + Je"'"-' /„ {mr)\ dm. (2) 



