688 ELECTRICAL METHODS [Chap. 10 



cedure is applied to both fields considered here, two figures eight result, 

 rotated 90° in respect to their larger axes. Amplitudes resulting from 

 each are superimposed. If in any given direction the amplitude of the 

 large field is PQ and that of the smaller field PR, the resulting amplitude 

 is P^ = PQ -{- j PR. Plottmg the amplitude of the PS vector for the 

 entire horizontal plane, we get a lemniscate. If the small out-of-phase 

 field were not present, a single figure eight would represent the amplitude 

 variation. An absolute zero would be obtained in the direction of the 

 equipotential line. With an interfering field, however, the sound never 

 vanishes. Only a minimum is observed, whose sharpness depends on the 

 amplitude of the out-of-phase field. 



Out-of-phase fields in electrical prospecting may be due to a variety of 

 causes. Currents traversing media of different conductivity, capacitance, 

 and inductance, will be shifted in phase. Further, ground currents and 

 electrode leads will induce out-of-phase currents in adjacent conductors. 



For two field components at an arbitrary angle with one another and 

 with a phase shift of 90°, polarization conditions are the same as for two 

 components at right angles to each other but with an arbitrary phase shift. 

 Two adjacent currents with a phase shift of 90° produce a transverse com- 

 ponent with a phase shift depending on the respective amplitudes of the 

 currents. If the one component is X and the other transverse component 

 is Y, we have, therefore, for the general case of two out-of-phase components 

 at right angles to each other: 



X = A sin wt 



Y = B sin {ut - ip), (10-24a) 



where co is the angular frequency and (p is the phase shift. Since sin cot = 

 X/A, and Y = B (sin ut cos <p — oos uit sin <p), Y becomes equal to 



B (r-cos <p - a/ I - ^-sin (p); or, 



Y - B-- cos(p = -B/l/l - ^2'^'^*^- 

 Squaring both sides and dividing by B" gives 



- - ;^ cos ^ + ^2 = s^^' 'P- (10-246) 



Dividing by sin ip, 



Y' _ 2XY cos ip X' ^ ^ 



B^ sin^ <p AB sin^ <p A^ sin^ <p ' 



