360 BELL SYSTEM TECHNICAL JOURNAL 



In a double summation, the result is independent of the order in 

 which the summations are taken. If then we write 



Vg = ^ e V C0S[9-*;„(2J_1)1^-P;,J(2J_1)^ (212) 



'"=l(2i = 2) 



2J=p 27rr^(2i) , ^ 



Wg = J^ ^-^~^^^oosW-<t>,n(2l)\-PM(2l)^ (213) 



2'=2(;„=1) 



m=Q 2?rr^(2i) 



yg = ^ e"-' V '^°^t^-*m(20lg--Pm(20_ (214) 



'"=1(2J=2) 



The expression for the total current may be written 



Jg = IgllgVg + Ig_„Weyg. (215) 



If the transducers in the circuits from each antenna of a pair are so 

 related that 



Pm{2l) — Pmi2l—1) = P c, (216) 



the expression for the total current becomes 



Je = UeVg[Ig + /fl_,e-^'2.s/V]cosS,-P.],-P._ (217) 



The directional diagram in terms of relative directional receptivity is 



Jo Wo ^0 



7, + V.6-^'^--/^'"-°-^6-^° 1 



Since there has been no assumption to this point of the character 

 of Ig, the significance of the coefficients ug and vg may be determined 

 by assuming 



(1) Ig = Iq, which is the directional characteristic of a vertical 

 antenna 



(2) 5 = 



(3) Pc= ^. 



Consideration of (218) in light of (211) and (212) under these con- 

 ditions leads to the conclusion that 



tig , Vg 



— and — 



Mo ^0 



