372 Dr. Balth. van der Pol on the Propagation of 



or when we introduce 6' as angular distance from antipodes, 

 i. e., 0' = 7r — 0, we have 



?*(-/*> 



\VTtJ -fix*; 



v-iv r> \vttJ —fix** -ixTT 

 . e 



I ICOSV7T y#' 



X (*" 



atf' 4-^/30' -^ .ae'-a^flO'+^X # . , ( l0 ) 



analogous to (5), but near 6 = ir the second term within the 

 brackets cannot be neglected as here the effect of the waves 

 which travelled over the longer distance ir-\-6 ] (i. e. the 

 other way round) is of importance. The effect of both waves, 

 1, coming round the arc 0, and 2, round the arc 2tt — 0, 

 interfere near the sender's antipodes and give rise to inter- 

 ference rings in the region near 6 = tt. 



The region in which (10) is valid is bounded by two con- 

 ditions : The zonal harmonic P _ L (—fi) may be replaced by 



the Bessel function of zero order when x(tt — 6) z = x9' z <^1. 

 The approximation subsequently introduced for J , however, 

 is only valid for x{ir — 6) = a;0'^l. It therefore follows that 

 (10), which conld further be derived at once from (5), has 

 only a very limited region of validity. 



In order to discuss the behaviour of the waves in the 

 immediate neighbourhood of the antipodes, |Er.| and | H^ | 

 must separately first be found. 



We have for this region : 



n=- * ^w ,/J oM^-^)t ; 



x h a P cos V7r 



in which series again the first term is predominant. The 

 magnetic and electric force therefore become, after omission 

 of terms only affecting the absolute phase : 



h^^V^k*-*)}. . • • (11) 



The electric force, however, is 



Vr^y-^Mvi-r-e)}. ■ . . (12) 



It is seen, as might have been expected, that near the 

 sender's antipodes the waves coming round " both " ways 

 interfere and produce interference zones. The magnetic 

 force in the neighbourhood of 0-^ir is again of the same 



