[ 105 ] 



IV. The Transmission of Electric Waves over the Surface of the Earth. 



By A. E. H. LOVE, F.R.S., Sedleian Professor of Natural Philosophy in the 



University of Oxford. 



Received December 19, 1914, Eead January 21, 1915. 



CONTENTS. 



Page 



1. Introduction and references 105 



2. Statement of the problem 106 



3. Simple limiting cases 107 



4. Analysis of the general problem 108 



5. Case of a plane boundary 109 



6, 7. Analytical solution for a spherical boundary Ill 



8. Historical Note 114 



9. Preparation of the series for summation 114 



10. Second Historical Note 116 



11. Transformation of the series representing the magnetic force 116 



12. Series representing the components of the electric force 117 



13-15. Approximate summation of the series 118 



16. Numerical results 121 



17. Criticism of a proposed alternative solution 123 



18. Approximate formula; 125 



19, 20. Experimental investigations 126 



21. Comparison with theory 127 



22. Discussion of the experimental evidence 127 



23. General conclusions 130 



1. EVER since the time, about 1902, when MARCONI first succeeded in sending wireless 

 signals across the Atlantic the question of explaining the mechanism of such 

 transmission has attracted attention among mathematicians. The question may be 

 put in the following form : The electric waves generated by the sending apparatus 

 differ from waves of light only by having a longer wave-length, which is, nevertheless 

 small compared with the radius of the earth ; and the curved surface of the earth 

 may therefore be expected to form a sort of shadow, effectively screening the 

 receiving apparatus at a distance. How, then, does it happen that in practice the 

 waves penetrate into the region of the shadow? Unfortunately, the question has 

 been investigated by different methods without adequate co-ordination, and the results 



VOL. CCXV. A 526. P [Published February 18, 1915. 



