ASYMPTOTIC DIPOLE RADIATION FORMULAS 671 



average values for earth but they vary so much with the locahty that 

 the diagrams can scarcely be regarded as giving more than a general 

 idea as to what may be expected of the formulas. 



The number attached to each curve is the height of its dipole in 

 quarter wave-lengths. 



Formulas (13), (14) and (19) are just what one would get by applying 

 the Reciprocal Theorem to two dipoles, one near the earth and the 

 other far away. The electric field acting on the one near the earth 

 is composed of a direct field and a reflected field which is Ri or Ro, 

 as the case may be, times the direct field. 



When the depth to groundwater, bedrock, an orebody or any 

 marked discontinuity in the electrical properties of the ground is 

 known and is not too great the effect of this discontinuity on the polar 

 diagram ought not to be ignored. The asymptotic formulas for any 

 stratified ground are got by putting the coefficients of reflection for 

 a plane wave reflected from the surface of that ground in place of the 

 corresponding coefficients in equations (13), (14) and (19). For a 

 number of rather obvious reasons it would usually be out of the 

 question to deal with more than one plane of discontinuity; one is 

 bad enough. The coefficients for a single plane of discontinuity at a 

 depth A are 



„ , ^9^ cos ds — viki'^ki^ — ki' sin- dz 

 Ri — 



and 



where 



ki cos d^ + rjiki'^k-r — /fer sin- 6. 



R, 



■yjko^ — kr sin- dz — r?2^] cos 6z 

 •V^2" — ^1" sin- dz — Tjoki cos dz 



_ /X2 1 + 5i _ /X2 1 + 5. 



Ull — 5i iJLll — 02 



g _ ^2^3 V^- k{' sin- dz - ks'fXi-^ - k{' sin'-' ^ ^2A\ ,..^-I:,^ ^i^ B. 



^2V3V^3^ — k{' sin- dz — ks-fJi-i-^k-/ — k{- sin- dz 

 and 



g^ _ M3-V^2^ - k;' Sin'^ dz - M2A/^3' - ki' sin^ ^,.,^^|^TZ^;7— T^ , 



At3'V^2^ — k{^ sin- dz + fJL2-^k.r — k{^ sin- dz 



If A is not large and ^3 is considerably different from ^2 then 771 

 and 7)2 will differ considerably from unity. 



