Electric Waves round a Large Sphere. 757 



providing that u is multiplied by 



1 i C 1 c 5 x 2 5 c 2 x 2 \ 



~*~Sz L(l_a? 9 j*"*(l-oV)* 3(^1-^)3 "3 (1-^.8)1 J ' 



or, in the present case of a large distance from the sphere, by 



1+M l —. + l * 2 , V • • (163) 



As before, 1 + *" J,X » becomes 2 +^ 2 /- 2: (l — ^ 2 )^j an( ^ with 

 these values, and 



R n =:(l-X 2 r% R*r=l, 



we find for the problem in hand, 





zv=?z{ \/l — # 2 -f#sin~*#J ^rhr-^-zxj), 



where 



,2 



i= | (I-* 2 )"* + ~(l-.^)-^ + ^ (1-*V*- (164) 



Differentiating, we havey?ns£ approximations to U\ and w 2 

 in the forms 



Ul =i(2-x 2 )(i~x 2 r*, 



u 2 =5 \zx (6 — /e 2 ) (1 — ■ x 2 ) ~* > 



These are, as before, not required to the second order. Let 

 C = cos</>, and S= sin 0, Then at the sero point, u being 

 its first approximation, 



Vw = (2-S 2 )/2S0 2 , 

 u 2 /^(6-S 2 )/40 4 , 

 X^(3 + 14S 2 )/24(J 3 , 

 * s *i/C, ^s = S/C 3 , ^=(l + 2S 2 )/0 5 , 



the values of the u's being quoted from (152) with cf> written 

 for0. 



Thus, on reduction, 



^ = __V^-^)-^^ + -^=(34-2S 2 )/12C 3 . (165) 

 u 4 \v 2 2 3 tyV v 2 z u v 2 u K ' v y 



Phil. Mag. S. 6. Vol. 24. No. 143. Nov. 1912. 3 D 



