Electric Waves round a Large Sphere. 171 



and is very small, containing an exponential of large real 

 negative argument. Thus 



yp =-2tti % v va v G(0)e- lV( P 



4sin 2 d ^ C" d$e~ lv * 



%va v 

 Jo 



ka 2 cW v v j e x/'2 ^(costf— cos<£)' 



and the important part of the integral is contributed near the 

 lower limit. Writing <£ = + f and neglecting square and 

 higher powers, the integral becomes 



and taking the leading term in the differentiation with 

 respect to 0, we finally obtain 



— 9 



yp = j^(2irsm0)^ v vta v e- ive +^, . (97) 



and it is now necessary to determine the residue a v . 



It is already obvious from the last equation that the poles 

 whose imaginary part has an order greater than zi are not 

 important, and they will henceforward be neglected, and the 

 summation restricted to those poles w T hich are of type 



it ; = z — izift, 



ft being of zero order and numerical. 

 Now near m = v, by definition of a„, 



■ ziK m (z)/~.z^K m (z)=-^, 

 7 dz v y m — v 



and therefore 



a v 



with m — v substituted after the differentiation. But v being 

 of the above form, the Bessel function is proportional to, by 

 results referred to earlier, 



~-Kf(p), 



where p = (m—z)(6/z)i, and f(p) does not otherwise contain 

 m or z. Thus "dfdm = ($/z)*7)/'dp i and moreover, so far as 

 the term of highest order is concerned, B/B^= — (6/*)«d/d/>. 

 For in this term, differentiation only lowers the order by zb, 



