Electromagnetic Waves round the Earth. 367 



When the sphere of radius a is supposed to have infinite 

 conductivity and the consequent boundary conditions are 

 taken into account together with the prescribed behaviour 

 of II near the oscillator, the solution is 



where 



w*)=a»«) | H« i («), 



H (2 J i being the second Hankel function which is zero for 

 infinite real argument. The ratio of the circumference of 

 the sphere to the wave-length, viz. ka, is moreover great in 

 comparison with unity. 



Now Watson transformed this series into a contour integral 

 which in turn he modifies into the rapidly converging series 



n M)= z-T 2 ~^> T,7^Z 



bah v rd£ s _Aka)- 



COS V7T ^ 



L 0$ 



(2) 



where the z/s are the (complex) roots of %\_i(ka). This 

 series is also valid when b = a, and the dipol is therefore 

 placed on the surface of the sphere as required in the 

 problem of Wireless Telegraphy. 



It is further shown that when x is written for ka 



4 — hiri 



where P P =i(%) i , 



f. being thejoth root of 



J_ |( |)_jj(f)=0. 

 Further, an approximation for 





is found to be 



2 



x* e 



mi — 1 



ox e p 



so that when the oscillator is placed on the earth's surface 



2 V 2 



