Electric Waves round a Large Sphere. 291 



"When sin 6 is not loo small, Hobson * has shown that 



p W!^ 



IT 



cos . (n-\r^)6 — — 

 (2 sin 0)* 



?J7T 



is cps.(n + j)0--j- ^ 



h o » -lu — /.> • ^ — + • • 1, 04cn 



2.2>i-M (2 sin 0)3 j 



the asymptotic expansion generally quoted being the first 

 term. We require the first two significant orders in m or 

 n-\~2* These may be obtained from the terms 



j— , in /cosl m6— t ) . cos(m0 -I- — V" K 



P^-v/ ^ (m4) V — '+ — — ^ — -* 



r,lW "Vtt tsr(m) 1 (2 sin 0)i 4m * (2 sin 0)1 j 



Now by Stirling's theorem, 



sr(m) = r m m w \/2W|i + ^ + • • 4 



and therefore to two significant orders in m~ l , 



-3r(?n) \??2/ \ 2??i/ 



The hyperbolic logarithm of this function may be expanded 

 in the form 



(l_ logm) _^ + _±_ + . . .) =; _ ^L + . . . _i l 0g m , 



and therefore 



-33-(m — -J) 



= w " i ( 1 -^)' 



-sr(m) 



so that the second approximation to the asymptotic value 

 of P»(ft) when ?i is large becomes 



PM Olf, 1\| /. ^\, C0S ("' g+g -T) l 

 P » = V ^i^l 1 -^)! 00 ^ "I) + - Smsinfl "J' 



which may be reduced to 



p » -v^5{-Kf) + t'4 rf -?)} (141) 



* PhiL Trans. 1806, A. 



