Electromagnetic Waves round the Earth. 371 



2ira being the circumference of the sphere representing the 

 earth, and therefore in the case of Wireless Telegraphy 

 equal to 4. 10 4 kilometres. 



If an error of 10 per cent, is allowed, only the first term 

 of this series can be retained for those values of 6 which 

 satisfy 



<9> 0-746.*"* = 0-0218\I M ., 



where X is measured in kilometres, or, which is the same 

 thing, from distances d (in kilometres) from the sending 

 station such that 



d K . M > i40i/x u . 



When an error of only 1 per cent, is allowed this condition 

 has to be replaced by 



<*„m. > 420 ^x K . M . 



For regions on the sphere nearer to the oscillating dipol 

 more terms of the series are required. Moreover, with the 

 first term only a fairly good agreement is already obtained 



at a distance # = 0*746<2? _ * with the formula directly derived 

 for the plane problem (Hertz, Abraham), viz., 



E 



■i -»<£)•» » 



where |E Z | is the electric force at right angles to the plane 

 boundary surface; here the factor 2 arises solely from the 

 reflexion due to the dipol being placed near or on the con- 

 ducting plane surface. The ratio of \E r \ calculated from 

 the first term of the spherical problem to the [ E | of the 



plane problem for = 0*746 #~* (corresponding to d KM = 



140 VX K . M .) is about 0*8. 



W r hen we now examine the electric and magnetic force 

 near the transmitter's antipodes, that is near = 7r, Laplace's 

 approximation for the zonal surface harmonic has to be 

 replaced by Mehler's 



Pw(cos©) = J {(?i-f i)o>}. 



Near = ir we have therefore 



cos vir /*** s/'hrviir'-d) . efl xl * 



>sIv(tt-0)- 7T 



