RADIATION CHARACTERISTICS OF AN ANTENNA. 205 



Returning now to equation (25), we shall integrate the third and 

 fourth terms, setting them first, and shall substitute (26) to (30) for 

 the other terms, obtaining 



2/2 A 2tv , , .) r sin2 B , sin^ 5 sin 2 ^ 



2P A 2ir ,^ J r sin'B . 



• 2A 



cos 2^ 1'+^ cos {2 Au)du sm2B T+i sin_C 

 "^ 4 J-i l + « "^ 4 J-i 1 



(2 Au) du 



^, \ r, C cos (An) du , . „ r+^ sm (Au) du ] 

 — cos G ' cos B I — - — , h sm /j / — ; — ; y 



J-\ \ + U J-\ \ + u 



. (31) 



Let us now write 



7 = 2 ^ (1 + u), 

 2 Au = y - 2A, 



^^' = 2 A' 



du _ dy 

 1 + w 7 ' 



then the second and third integrals of (31) become 



cos 2 B r+i cos (2 Au) du , sm2 B r+^ sin (2 Au) du 



4 



cos 2 5 



r+i cos (2 Au) du sm2B T+i sin (2 Au) 



J-\ 1 + M 4 J-l 1 + w 



{ COS 7 COS 2 ^4 + sin 7 sin 2 A] — 

 o 7 



, sin 2 5 ^-4 • o ^ I '^ 'i' 



H r — / sm 7 cos 2 A — cos 7 sin 2 ^ } 



4 Jo 7 



cos (2^ + 25) p^ cos 7 (^7 sin (2^ + 2^) p-^ gin 7 c ?7 



Jo 1 4: Jo y 



4 

 cos 2 G C'^^ cos 7 , , sin 2 6' f'^'^ sin 7 



In like manner, the last line of (31) becomes 



dy . 



cos2 G r^^ ^^^ dy - cos 6* sin G f^'' ^^^ rf7. (33) 

 J y Jo y 



