MAGNETIC METHODS 197 



The horizontal component at P is 



dVx 2thl 



a/ -d^ + p (^^"^ 



The second effect (due to magnetization of the layer by the field at 

 right angles to its surface) may be obtained as follows : Let 1 2 and — 1 2 

 be the densities of magnetic charge on the two surfaces of the layer. The 

 potential at the point P due to a magnetized strip which is perpen- 

 dicular to the plane of the paper and has a thickness t is 



2tlz cos 9 

 r 



(Compare Equation 80.) The potential Vz due to the magnetized surface 

 may be obtained by integrating the potential due to the magnetized strip 

 with respect to x ; that is, 



Vo = 



ypdx 



yo^+ {x — xq)' 



L yo 3'o J 



The X and the y components of the magnetic field at P are 



r yp yo 1 



KA^2ut^_^ yjL 



dxo 

 and 



9^2 or X r •*'0 1 -^0 



9yo 





SPECIAL CASE 



Suppose that one end of the layer is very far removed from the earth's 

 surface ; that is, suppose that Xi is very large. For this case, the equation 

 given above for the potential due to cross magnetization reduces to 



V2 = 2hJt^n--'^+Y'] 



The X and 3; components of the anomaly due to the cross magnetization for 

 this case are : 



9yo yo^ + ^0^ 



* This equation is obtained by differentiating V3 with respect to :vo under the integral 

 sign and then integrating the resultant expression with respect to Xo between the limits 

 xo and ;ri-,ro. In carrying out the integration use is made of the formula 



i 2vo^ dx __ x — Xo i dx 



)[yo'+ix-xoyy~ yo''+ (x-xoV ;y+ ix-xoV 



