Electric Waves round a Large Sphere, 293 



approximations, and multiply the non-oscillatory factors by 



l + lfin — lfinr Or 



i_ f 1_ _ c 5 g _5 cV \ 



• »*\(l-«»)* (l-cV)± 3(l-« 2 jt 3(l-cV 2 )fJ ' <■ 147 ' 



Finally , % w is required. In the first approximation in the 

 region of brightness, we wrote ^=0. The more accurate 

 value is given by (27) as 



Xi=«»/2*(1 -«»)!, 



so that to the same order, 



l + ^-^l+g^}. . . . (148) 



Using finally the asymptotic formula for the zonal harmonic 

 and the values of the various functions derived in this section, 

 the series to be summed may be regarded, to the necessary 

 order, as identical with 



S = 2 ue lZ \ 

 where 



zv = (j) n -cj) nr -md (H9) 



with the same value and derivates as in the first approxi- 

 mation, and where by (1 10), and (143) et sec]. 



ix sin 2 6 



u = — - 



ka 2 



x 



(RJU)* (l -r *A-*Ar) (l + «*«») 



/ ,r# W 3*cot#\ _^ 



te^j 1 1 — 8^/ 4 

 = (i-^.i-^t l 1 + - g > ( 150 > 



provided that 



2& sin <9 i ff 

 — : •**> 



3 cot (9 1 r 



2(i-^ 2 )f 8# 8 LCi-* 8 ; 1 (i-c 2 .^ 



+ 3(I^^^3(T^V)^ 



