Electric Waves round a Large Sphere. 295 



The sum is by (138), 



= ^2tksin 2 0e- im {l+ l ^\, . . . (155) 



= -2ckm^6e- lKK ^ 1+ " 

 where 



S = \^/*/w:=(3-h8sm 2 <9)/12cos 3 0. . . (156) 



Thus, superposed on the effect of the oscillator in the presence 

 of a plane reflector, there is an additional vibration o£ relative 

 order z~ l for which the magnetic force is 



7 = -i-. sec 3 5> sin 0(Zi-8 sin 2 0)—^, , . (157) 



when points in the region of transition and points close to 

 the axis are excluded. This corresponds to an oscillator 

 which, when undisturbed, gives a magnetic force 



The resulting amplitude at any point in the region is only 

 altered to an order z~ 2 by taking account of this vibration. 

 For we may write 



i ih /, S 2 \^ 

 1 + -= ( 1 + -J e* ; 



z 



-0+5/ 



but this approximation does not determine the amplitude, for 

 terms of relative order z~ 2 have already been neglected. A 

 determination of the second order terms could, however, 

 readily be made. The phase at a distant point is changed 

 from JcR to JcR — Bjz. When the first approximation was 

 determined, it was stated that in a case in which /ca = 10 6 , an 

 error of relative order 10~ 12 only was involved in the assump- 

 tion that the region of brightness is determined by the plane 

 reflector effect. This statement is now justified. 



The corresponding problem for points near the axis requires 

 a different treament, and will be investigated later. It will 

 be shown that, as in the first approximation, the type of 

 solution does not change near the axis. 



