PROPAGATION OF PERIODIC CURRENTS 535 



is due to the distributed impressed voltage acting in the leakage 

 admittance from the wave antenna to ground (this leakage admittance 

 being regarded as uniformly distributed). By substituting the values 

 f{y) and F{y) given by (110) and (111) when x is replaced by y, then 

 carrying out the indicated integrations, and finally transforming the 

 results somewhat, the constituents corresponding to (50), (51), and 

 (52) are found to have the following formulas: 



_ 5/(0, d) cos 6 sinh [(t - r cos ^)5/2] ^_,^^^_,,,„ 

 -^^ ' ^ 2K (y-Tcosd)s/2 ' ^^^^^ 



Ms, 6) = -:^^e-^^ (113) 



Ms,d) =^^^^^-r^eos.. (114) 



The fourth constituent, Josis, d), corresponding to (53), will be 

 omitted, because it is relatively unimportant and also because its 

 formula is found to be somewhat lengthy. 



The valuable directional selectivity of a wave antenna resides 

 mainly in the directional properties of the admittance j(-^) ^)A/(0, 0') 

 whose value is found by dividing equation (112) through by 5/(0, 6'), 

 where 6' denotes some fixed value of 6 (usually 6' = 0). This ratio 

 may properly be termed a 'directional admittance.' The correspond- 

 ing admittances obtained by dividing (113) and (114) through by 

 5/(0, 9') are not usefully directional, the former being entirely non- 

 directional, and the latter only directional as regards its phase angle — 

 not as regards its absolute value. By suitable choice of the length of 

 the wave antenna, the constituent represented by (112) can be made 

 to have high directional selectivity, while the constituents correspond- 

 ing to (113) and (114) become relatively unimportant (except over a 

 few narrow ranges of the incidence angle d).-'^ 



A Crosstalk Problem 



This problem is concerned with the derivation of formulas for the 

 first-order crosstalk between two simple open-wire telephone circuits 

 of which one is non-transposed and the other is once-transposed, as 

 represented in plan view by Fig. 13. 



The once-transposed circuit is taken as the primary, and the non- 

 transposed as the secondary. Each extends from x = to x = s; 

 and the primary is transposed at its mid-point x = s/I. 



^^ For a detailed study of the wave antenna, the reader is referred to the well- 

 known paper by Beverage, Rice, and Kellog entitled 'The Wave Antenna' in 

 J. A. I. E. E. beginning with March, 1923. 



35 



