4 L. J. PETERS AND J. BARDEEN 
potential. Any changes of resistivity in the subsurface beds alter this normal 
potential. Thus measurements of potential on the surface of the earth give 
an indication of these changes of resistivity. Most prospecting methods differ 
only in the spacing of the electrodes and in the manner in which the potential 
is measured. The same mathematical theory thus applies to all of them. 
The theory is simplified by the fact that problems of steady flow in con- 
tinuous media are the same mathematically as similar electrostatic problems. 
A current J entering an earth of conductivity o atan electrode A may be re- 
placed by a charge of g=J/2m0 at A. The potential at any point in the earth, 
due to this electrode is V = J/27ar, where r is the distance from the electrode. 
If several electrodes A, are on the surface of the earth with currents J, 
entering the earth, the potential at any point P in the earth is 
1 Me 
aia aa (1) 
Hsp i te 
where 7, is the distance from P to A,. It may be noted that algebraic values 
of the current must be used in this sum. Thus the potential at P in Fig. 1 
for the current flow between two electrodes is 
I 1 1 
V =—(— -— (2) 
2x0 T1 1%? 
This additive property of the potentials is general and is not confined to a 
homogeneous earth. Thus we may simplify the theoretical discussion by con- 
sidering the potentials due to a single electrode. 
In order to illustrate the type of departure from normal potential that 
may be expected, a simple problem will be solved. The earth is of conductivity 
01, to a depth a and is of conductivity o2, below this depth. A current J flows 
A r 
P o- 
CMM MM; é; 
r 
Fig. 2. Coordinate system for two-layer problem. 
into the earth at the origin of cylindrical coordinates r, z, 6. (See Fig. 2.) Let 
the potential in the surface layer be V; and the potential for z>a be Ve. The 
conditions to be satisfied by the potential are: 
(1) The differential equations expressing the continuity of the currents 
a4Vy) DOV y. 4 .07V iy 
yy Sy eae HS, Sa ey (3.1) 
or? for Oz? 
V?V;2 — 0. (3.2) 
(2) The continuity of current and potential at the boundary 
Vi=V:atz=a (3.3) 
148 
