RADIATION CHARACTERISTICS OF AN ANTENNA. 203 



2/2 ., ( 27r 



p = cos- < — 



c 



J 27r , , J r 2D C'^^ COS^ (^ COS g) dd 



, . , „ r-/2 cos2 d sin2 (^ cos 0) c^9 , , ^ r-/2 ( 



+ sin- B I ^ — h cos^ G I — 



Jo sin t> Jo SI 



i 



— 2 sin j5 cos 5 



dB_ 

 sin 



''^/^ cos d sin (^-1 cos 0) cos (A cos 0) c?0 



sin0 



n I '''^ COS (^ COS ^) (^ 



2 COS B cos (j ' 



*Jo 



sin 



. o • o ^ p/2 cos sin (.4 cos 0) rf^n .^.. 



+ 2 sin 5 cos G I ^-T ^ — • (24) 



Jo sin Q J 



This equation when integrated gives the power radiated from the 

 vertical part of the antenna. The integration is a tedious operation, 

 and is given in the next section, which may be omitted by readers 

 not interested in the mathematical processes involved. The result 

 of the integration is found in Section 10. 



9. The Integration of Equation (24) . — By the use of such trigo- 

 nometric equations as 



1 + cos 2x 



cos^ X = 



sin- X 



2 



1 — cos 2x 



2 



the squares of sines and cosines in the integrands of (24) may be 

 avoided, and equation (24) written 



2/2 J 27r ,^ J r/1 , _A cl de 



, cos 2 5 r'^/2 cos (2 A cos d) dB sin^ B C"^'^ • . ,. 



H 7y — / • r—. ^— / sin d dd 



2 Jo sin0 2 Jo 



H ^— / sin d cos (2 A cos 6) dd 



- Jo 



sin 2 £ r^/'^ cos sin (2 A cos 0) (/0 



2 Jo sin 



^/2 cos (.4 cos d) dd 



— 2 cos B cos G' 



*Jo 



sin 



I .-. • n /-( r COS sin (A cos &) c?0n ,^_^ 



-+- 2 sm ii cos G I -. — r • (25) 



Jo sin 9 J 



