194 



EXPLORATION GEOPHYSICS 



diamagnetic. On the lower surface of C the surface density of magne- 

 tism is — /s. On the upper surface of A the density of magnetism is /i = 

 kiZ and along the upper surface of B it is I2 = k^Z. This condition is 

 p equivalent to the following: Density 



'^*-.,,,.,^^ Iz on upper surface of C, density 



Ldbf- ^1 ~ ^3 °" plane CA, density I2 — h 



■::i:-: -'-.-•-,'- /r-.: •.--.-. .^v -'-.-. ^rr^-T-prrrr-. on plane CB.^ 



field in the direction of r is computed 



Fig. 85. — Sketch showing quantities which , i,-,,- t ir to-t- 



enter into the calculation of the effect of by SUbstltutmg dx dy for dS in Equa- 

 upper surface of C. ^j^^^ g2 . ^hat is, 



3 dxdy 



AZr = 



SI 



If the height of the measuring instrument above the surface of the 

 ground is designated by 8, the vertical component AZi may be written 



AZi = 



/ 00 / 00 



1 I ^3 cos e 



) ) f 



/ _ 00 / — 00 



dxdy 



= hB 



/cO I— / 00 

 / — 00 I— >/ 00 



dx 



-Xir+iy-y^y + 8^T 



dy 



To integrate the quantity within the brackets, set (y — yi)^+ 8~ = a~ and 

 introduce a variable <^ defined by the relation x = a tan 4>. Thus 



1 



dx 



[(x-xiy+iy-yry + B^V 



- 00 



■t/2 



i 



dx 



i 



[(x-xiy + a^y^' 



■ 00 



2 



^cos<^rf<^-^--^_^^^,^g. 



-5r/2 

 38 



/ 00 / 00 



\ \ [(x-x^y+i 

 / — 00 / — 00 



{y-yi)' + 8^r' 



dxdy = I3 8] 



{y-yiy + 8' 



dy 



= 



27.8 



tan" 



.^ y-yi 



T =27r 



J — 00 



That is, the vertical component AZi due to the magnetic density on the 

 upper surface of C is ZttIs. 



* This is an approximate treatment which does not hold strictly at the boundary 

 AC and BC. 



** For simplicity, the illustration shows a vertical section through a point P 

 i^i, yi, S). 



