370 l)r. Balth. van der Pol on the Propagation of 



and by simple differentiation, bearing in mind that <vp\ 



and that the time-factor e lU)t was everywhere omitted, we 

 arrive at the approximate expressions 



(?) 



and 



13 



<P\~ ca\~dd\ = a 3 v/sin 6 p* 



13 



-nirv^iV • " : ;.'■-'. ' (; 



which shows that the same expression is obtained for | H J 

 and |E r |. 



As of the series (7) and (8) the first term is by far 

 predominant over the greater range of 6, the expression for 

 the wave-amplitude takes the form 



a — u e 



where A and u are independent of 6. 



1 his form of solution is consistent with the following physical 



interpretation: — The first factor I ~r— — A shows that the 



waves follow closely the sphere's curvature, with a divergence iri 

 accordance with the a> ea of the sphere over which they spread, 

 the radius of the parallel circles being proportional to sin 6 ; 

 but, while travelling, they continually give off energy at a con- 

 stant rate to the surrounding medium as shown by the second 



factor (e~ ue ) ; 



The equality of | E r j and | H^ | points to the fact, as was to 

 be expected, that at the surface of the sphere the waves 

 behave like plane ones, and the calculation of the amount of 

 energy absorbed in a receiving system may be carried out 

 as if the waves were really plane. 



For wave-lengths X not too long and angular distances 6 

 not too small, the series still occurring in (7) and (8) is seen 

 to converge very rapidly. In fact we have 



P 



1 -07000 a;*0 1 -2 232^0 -1 -3'82»*0 , 

 0-8083* + 2-577* * 3'83 ' 



i 2wa. 



where ,v= > 



A, 



