Chap. lOJ ELECTRICAL METHODS 789 



that B declines in the same proportion, so that B = Bmax. djr. Since 

 Bmax. = A, B = A cos 0, so that by substitution of the above and of tan Q = 

 y/d in eq. (10-53c?), 



tan 2,A = - o^'''1^^2. 1 - (10-54«) 



cos 26 — tan^ 6 — I 



The axes of the elhpse follow by substituting B = A cos ^ in eq. (10-53e) 

 and by dividing the result by B : 



a^ b' = '^ . (10-546) 



2 + tan^ d + — r- Vl - 2 cos^ ^ - 3 cos^ 

 cos^ 6 



Further relations for elliptical polarization may be developed by con- 

 sidering different surface variations of the quadrature fields and different 

 ratios of their maximum amplitude to the loop field. 



(e) Theory of interpretation. Although, for complete definition, the 

 electromagnetic field theoretically requires six quantities (three compo- 

 nents and their phases), it is seldom necessary to determine all of them. 

 If the primary loop has been laid out parallel with the strike, the X com- 

 ponent is generally negligible, and measurements are concerned with the 

 in-phase and quadrature constituents of the Y and Z components only. 

 Of these, the vertical and horizontal quadrature components are of greatest 

 diagnostic value. 



From the theoretical relations given below for the out-of-phase fields 

 produced by various geologic bodies, the requisite formulas for any other 

 electromagnetic method not directly measuring these components may 

 be deduced. For instance, the horizontal and vertical direction of the 

 field determined wth a simple induction coil follows from the character- 

 istics of the ellipse of polarization, that is, by compounding the in-phase 

 loop field with the quadrature field of subsurface bodies. Intensity ratios 

 and phase differences (as determined by ratiometers) may be calculated 

 by compounding the loop field with the subsurface fields and their correct 

 phases for successive points.^* 



Interpretation of results obtained by inductive-electromagnetic methods 

 in mining differs greatly from procedures applied in the investigation of 

 stratified ground. Absolute values for the magnetic fields of subsurface 

 currents may be calculated if the strength of the induced current is known. 

 It is difficult to determine these currents theoretically. However, since 

 most inductive procedures measure the magnetic field relatively, it is suffi- 



"^ If long grounded cables are used, the observed fields result from a combination 

 of in-phase components (due to conduction) with quadrature fields (due to induc- 

 tion), and the phase shifts are calculated accordingly. 



