RADIATION CHARACTERISTICS OF AN ANTENNA. 



229 



P /■V/2 i 



= I dd sin (A cos 6) < cos B cos {A cos d) — 



CTT Jo I 



sin B cos sin (^ cos 6) — cos 6' 



J 



\ 



Jf^^' cos <^ £ 

 o 1 



COS 6 



cos 6 sin ^ sin B 



sin (fi sin cos </>) c^ 



Jo 1 



sin^ d cos^ <^ 

 cos (/0 



sin^ cos^ (/>_ 



(89) 



This is a very complicated expression involving the integral of an 

 integral. 



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



Let V 



1 



'-'' cos </) sin (B sin d cos 0) rf0 

 1 — sin^ 6 cos^ 



(90) 



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



v=r+ r. 



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

 replacing <^ by 0' + tt, we find that the integrand is unchanged, while 

 the limits become and tt, so we mav write 



X 



^ COS sin {B sin Q cos (^) d^ 

 1 — sin^ B cos^ 



(91) 



Again decomposing this into the sum of two integrals we have 



nyo fJir/'Z 



I 



r/2 \ 



(92) 



and changing the variable in the second integral by putting 

 = TT — </)', the second integral becomes 



f ' = r 



— d4)' (— cos (^') (— sin {B sin cos <p') 

 1 - sin2 B cos2 </)' ' 



