KADIATION CHARACTERISTICS OF AN ANTENNA. 213 



along the .r-axis instead of the z-axis. This arrangement is shown in 

 Figure 8. The plane of the zero value of S is now to be fixed as the 

 plane of the z and z-axes. 



Now using the current distribution of equation (1), we must replace 

 / by a + x', which gives, when treated as (12) was treated, 



•; 'IttcI { 2Tr f, .r'2 - 2x'x -2za-{-a'\ } 

 '^^ = ^T- ^°M V V'' - '^ 2io ) \ 



sin j -^f^ - a - x'j ' dx'. (48)' 



The fictitive current at P" is just equal and opposite to that at P', 

 with, however, a different distance from the point P, so we may write 



y 2c7r7 



J2 = 7 cos 



JT(-^-'-°- "'"n:""^'" )} 



sin ] ^(^ - „ - ..') [ <!.,'. (49). 



Whence by addition, employing the trigonometric relation 



cos (a + j8) — cos (a — /3) = — 2 sin a sin |8, 

 equation (47) becomes 



^j? m 47r/ sin iA . j 27r/ x'^ - 2xx' + a?\ } 



clL^ = dH^ = sm -^ ^— ct — ro r ' V 



'ocX (XV 2ro J ) 



. 2'iraz . i 27r/Xo ,\ } ' 



.^^"-x^^^Mw'^'^Jr". 



In this equation we may as usual replace — — by A. Also we may 



A 



make an approximation as follows : For large values of 



To 



X- - 2xx' + a- 



XX 



= — — = — x' cos \p. 



2ro ro 



' • • . . x'~ -\- d^ 



In making this approximation the neglected term is — — — , and' 



2ro 



