GAIN OF DIRECTIVE ANTENNAS 85 



Similarly the time phase of any particular element relative to the 



origin is 



8nNN = Lin - l)b + (N - \)B + (N - l)5]r. (2) 



The instantaneous value of the electric field at any remote point P 

 due to one of these sources is given by 



It 



En' = A cos — (C/ — Rn') -^ 8n' = A COS \l/n', (3) 



where n' = nN. 



The resultant interfering effect at a point P due to n' such sources 

 all of equal amplitude is given by 



£2 = „'£o2 + 2£o-[cos (^/'x- 1^2)+ cos (^1- 1^3)+ cos (4^i-rPd + -- • etc. 



+ cos (\l/2 - h) + COS (\p2 - h) + COS (l/'2 - V^s) H CtC. 



+ COS (1^3 — ^i) + COS {\p3 — ib)-\ etc. 



+ COS {rPn'-i-Ml. (4) 



The summation above gives rise to three series as follows: 



•S'l = (w — 1) cos 27r(a sin 6 • sin (j) -\- b) 



+ (w — 2) cos 2-2x(a sin 6 - sin + 6) 



+ (w — 3) cos 3-27r(a sin d • sin cf) -{- b) + • • • 



+ cos (w — l)-27r(fl sin d • sin 4> -\- b), (5) 



Sy = (N - 1) cos 2ir{A sin O- cos </> + 5) 



+ (iV - 2) cos 2-27r(^ sin 6 - cos + 5) 



+ (N - 3)cos3-2Tr{As'md • cos + 5) + • • • 



+ cos (N - 1)-2t{A sin 6 - cos + B), (6) 



S, = {N - 1) cos IwiA cos 6 + B) 



+ (// - 2) cos 2-2t(A cos + 5) 



+ (iV - 3)cos3-27r(^cos0 + 5) + ••• 



+ cos (A^ - 1) •2Tr{A cos 9 -{- B), (7) 



such that 



£2 = £o2(^ _j_ 25,) (A^ + 25,) (A^ + 2S,). (8) 



Each series is of the type 



S = (ti — 1) cos X -\- (n — 2) cos 2x 



+ (» - 3) cos 3;c + • • • + cos (w - l)x-' (9) 



