Electric Waves round a Large Sphere. 287 



where 



A=r 3 /6r 2 



B = (3i^ 4 -^ 2 )/72r 2 2 , 



and where it has been found convenient to denote v (n) by v n . 

 Again, we may at once derive 



= (2r 2 )-i|l + ay + /37/ 2 + ...|, (126) 



a= — v 3 /3i' 2 



/5 = (ll^-9r 2 r 4 )/72^ 2 . 



where 



a=— 1? 3 /3» 



Thus 



V=U»/»'=V '+yV 1 '4-y»V a f + ..., 

 where on reduction from (125) and (126), 



V 2 - L (2 l2 ) . | f _ _ _ ^- _ - - + _ _ | . (127) 



In addition, we have by Maclaurin's theorem, 

 V '=(V) =(V) =v 



v/scav/a*),, 2\v=o 2 v/s.^) . 



Again, by (125), 



0»/3*)o = (%/2) », Q V3* ,2 )o = 2 A(». 2 /2) *, 

 whence 



•'9*/9< a ) = (2/r 2 )*, O 2 . E /3ft) 2 ) = -4A/«,= -2» 3 /3» 3 s , 



and 



v = i^L\ / <^ ^ 5!!/ 5fV I 



2 W A ~ 1 3* B» 2 + B.<-' 2 V W J o 

 2U„ 



,,,_■ IU," t., TV 1 .o. 5 r/| , 

 ( ^ 2 ' ' I T7 - * U - 4 v, + 12 7-7 ( ' (1 - 



8) 



whereas 



V»=U (2r L ,)-S (129) 



