Electromagnetic Waves round the Eartli. 369 



cos vtt is further very approximately equal to 



cos^7r=ie/ 3 ^ 7r + z>7r . 



Further, for all values of not near or 7r, that is for the 

 whole sphere except the zones near the sender and its 

 antipodes, Laplace's approximation for the zonal harmonic 

 P _,(—/*) of complex order yields 



(-1 



2 



(Id \ 



2v/sin# 



+ e 



}, • • w 



and the second term within the brackets can be neglected 



unless 6 is so near to it that e ^ x ^~ ' is comparable with 

 unity. 



The neglect of this second term amounts physically to 

 the neglect of that part contributed to the total value of 

 the potential function (or the wave amplitude) which arises 

 from the wave that has travelled round the angle (2tt— 6), 

 and therefore started in "opposite" direction from the 

 sender, passing on its way the sender's antipodes. That this 

 amount contributed to the total value of II is negligible, 

 except for points very near the antipodes (0~7r), is obvious. 

 When those points are for the time being excluded from 

 consideration (which is already necessary as for this region 

 the approximation used for P _ x is not valid), 



is found to be 



COS V7T 



2 



-) 



v sin 6 



^-i a+ D (5 ) 



Hence the expression for the Hertzian potential function II 

 becomes 





(6) 



