RADIATION CHARACTERISTICS OF AN ANTENNA. 



229 



/2 /V/2 _ ( 



= ~ I dd sin {A cos 9) \ cos fi cos {A cos 9) — 



CTT Jo ( 



sin 5 cos 6 sin (yl cos 0) — cos G 



cos 



r^'^ cos (f> sin (fi s 

 Jo 1 — sin^ 



sin cos (j)) dcf) 



cos sin 6 sin 5 



Jo 1 - 



2 ^ C0S2 <i> 

 COS (/) (/(/> 



sin^ d cos^ <A 



(89) 



This is a very complicated expression involving the integral of an 

 integral. 



We shall first proceed to perform the integration with respect to (f). 



Let I' 



-1 



^'^ cos 4) sin (B sin 6 cos <^) dcj) 

 1 — sin^ ^ cos^ 4> 



(90) 



and break the integral into the sum of two integrals thus: 



r= f'+ r. 



tj o ^'T 



By a change of variable in the second of these two integrals by 

 replacing 4> by (/>' + tt, we find that the integrand is unchanged, while 

 the limits become and x, so we may write 



-X' 



cos 4> sin {B sin 6 cos 4>) d4> 

 1 — sin^ d cos^ 4> 



(91) 



Again decomposing this into the sum of two integrals we have 



''iJo ^ X/2 \ 



(92) 



and changing the variable in the second integral by putting 

 (p = TT — (p', the second integral becomes 



t/7r/2 J7r/2 



— d(}>' (— cos <t)') (— sin (5 sin 6 cos 0') 



ir,2 a „^^2 J,/ 



1 — sm^ d cos^ (^ 



