200 



PIERCE. 



Eg = Ha = 



2ir I sin 9 

 Xcro 



r 



cos — (cf 



To — Z COS f^) 



27r/Xo 



Sin 



(^ - .') rf.'. (14) 



By reflection from the earth, which we shall regard as a perfect 

 reflector, we have intensities that must be added to the above. These 

 intensities may be obtained by considering the radiation to come from 

 an image point at a distance z' below the surface. The effect of this 

 is obtained by changing the sign of the z' in the cosine term of equation 

 (14), but as was pointed out in section 4 the sign of z' in the sine term 

 must remain. We obtain thus for the intensities due to the reflected 

 wave emitted by the vertical portion of the antenna the value 



Eg = H^ = 



2ir I sin 9 

 Xcro 



r 



t/o 



27r 

 cos Y" (c^ 



A 



To + z' cos 6) • 



sin y f ^ - z' ) d^'- (15) 



Adding the equation (15) for the reflected intensities to the direct 

 intensities of (14), remembering that if A and B are any two angles 



cos {A — B) -\- cos {A -{- B) = 2 cos A cos A, 

 we obtain for the total intensities at P the equation 



(16) 



„ „ Air I sm 6 



Eg = Ha = ; COS 



Xcro 



which resolves into 



— {d — To) f cos I -r— cos 



sin'^(^-z']dz', (17) 



Eg = Ha = - 



47r / sm d 2ir , ^ 



— cos ^— \ct 



Xcro X 



. ttXo T" 



ttXo r« 



- ro) [ 

 2Trz' cos 9 



cos 



2wz' 

 X 



2Trz' 



2irz' cos 9 . 

 cos r sm — ^ 



A A 



dz' 

 dz'l 



(18) 



