RADIATION 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 x and z-axes. 



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

 I by a-\- x', which gives, w^hen treated as (12) was treated, 



2iTcI 



j 27r 

 \ 27r/Xo 



cos -^ ^1 (ct — ;-o 



x'^ - 2x'x - 2za + an | 



2/0 



'^" I TU ~ " " ^ 



•'^ I 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 



2c7r7 ( 27r / .i-'2 - 2xx' + 22a + a^ \ ) 



. i 27r/Xo 



x'] |- dx'. 



(49). 



Whence by addition, employing the trigonometric relation 



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

 equation (47) becomes 



4x7 sin \}/ 



dE, 



^ = dHx = — — — -v^^~ sin < ^— ( ct — ro 

 /•ocX ( X 



27ra3 . 



sin ^;^ sm , , V . 



Xro I X V4 



j 2. ^^^ _ ^ _ ^-^ - 2..- + g^ 



2ro 



a — X 



27ra 



( 



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



make an approximation as follows: For large values of ro 

 x'^ — 2xx' + a- 



XX 



2ro 



= — "^- = — x' cos 4/. 



ro 



x" + or 

 In making this approximation the neglected term is — , and 



27'o 



