218 PIERCE. 



have called the power radiated from the vertical part of the antenna. 

 We shall call the second term above (56), when properly integrated, 

 the poioer radiated from the flat-top. The third term, since it coijtains 

 both sets of coordinates, may be called power radiated mutually. 

 These designations are merely for convenience in paragraphing the 

 mathematics involved. 



15. Power Radiated from the Flat-top. — Let us now enter 

 upon a determination of the power contributed by the second term of 

 the right hand side of equation (56), and integrate this term over the 

 aerial hemisphere; that is, the hemisphere above the surface of the 

 earth regarded as a plane. 



The element of area of this hemisphere is 



dS = r^ sin i/^ d^p d^. (57) 



This is to be substituted in the required term involving E^ and H^; 

 but these quantities involve the coordinate z, which must be replaced 

 by its value in polar coordinates 



z = ro sin i^ cos S. (58) 



Besides (57) and (58) we are also to substitute the values of E^ and 

 H^ from (55) into the term 



dp = -^^ (e^ H^ dS. (59) 



E^ and H^ are identical, by (55); the product will give certain 

 terms involving sinV, other terms involving cosV, and still other terms 

 involving sin t cos t; where t has the value given in (51). If we take 

 the time average for a complete cycle, or, if we prefer, for a time that 

 is large in comparison with a complete period, we have 



av. sin^r = av. cos^r = |; 



while the average of the product 



av. sin r cos t = 0. 



The integral form of (59) then becomes, if ^ = the time average 

 of radiated power, 



