ELECTRICAL METHODS 615 



field is in the 3; direction.* Assume also that the two fields have the same 

 frequency {w/lir) and that they have a time phase difference of ir + 6. 

 Corresponding to these assumptions, the primary and secondary fields are 

 given by the relations : 



X = Xq cos wt 



Y=-YoCOs(wt + e) 



where X is the field component in the x direction and Y the component 

 in the y direction, and Xq and Fo the amplitudes, i.e., the maximum values 

 of X and Y. 



These two equations are the parametric equations of an ellipse as will 

 be evident from the following analysis. On eliminating the parameter / 

 between the two equations and simplifying, one obtains 



72 2XFcosg X^ ^ ..- 



Yo^sin^e XoYos'm^e Zo" sin^ 



Equation 17 is of the form 



AY^ + 2BXY + CX^ = 1 

 where 



A = l/Yo^sm^d 



B = cose/XoYosm^6 



C=l/Yo^sm^e 



Hence (17) is the equation of an ellipse.** 



The analysis given in the above paragraphs treated a special case 

 of two fields oriented in space at right angles to one another. However, 

 it can be shown that irrespective of the number of secondary fields and 

 irrespective of their amplitudes, phases and directions, the resultant of 

 the primary and secondary fields may be represented by a single vector 

 whose tip periodically traces out an ellipse, t 



Determinations of the ratio of the major to the minor axis of the 

 ellipse at various stations in a given area will frequently give valuable 

 information on subsurface irregularities in conductivity. Such determina- 

 tions are conveniently accomplished by the double coil method. 



* Note that this assumption corresponds to assuming that a = ir/2. 

 ** It is readily verified that Equation 17 degenerates into the equation of a straight 

 line when 6 =:0. For on setting e = in Equation 17 one obtains 



y 2XY , X' ^ 



A.01 ■'^O 



or 



(XoY + XYoy = 

 and 



XoY + XYo = 



which is the equation of a straight line. 



t A. B. Eroughton Edge and T. H. Laby, Geophysical Prospecting, p. 278 (Cambr. Univ. 

 Press. 1831> 



