Electric Waves round a Large Sphere. G3 



In the paper in question, a portion of the proof assumes 

 that m is real, but the extension to complex values whose 

 real part is large and positive may be shown to be lawful. 



By reference to the original definition of K m (z), namely, 



v J 2 sin mir L J 



it may be shown that 



K»(,)4g)Vw/rQ(i + ,-^ + i;r(|)(i +e ^) 



+ 2-i r ©( i+eir ) + -}- • ( io3 > 



Thus the equation 



■d/-dz.^K m (z) = 

 becomes 



where 



^)=r(|)( ]+ ,-^) +r ^r(?)(i + ^) + ... 



But 



-dfdz . zty(p) = zty / (p) 'dp[bz-i*-htr(p) 



= -Z-Hir'(p)6hz% + ±f(p)} 7 



since for a value of p of order unity, ~dp/"dz = — (6/z)i, as in 

 a previous section. The first term of the last expression is 

 therefore preponderant, and the equation for the required 

 zero may be treated as 



to a sufficient order for the present purpose. We must 

 therefore solve the equation 



(i+^Wl) + (i-.-')r(i)i; + ( 1+ ^)r(De; 



certain of the terms vanishing. This may be written in the 

 form 



+ OKI + 



I) V 3 3! + 3 2 6! + ••/ 



2! -3 5! - r(i) 



. . . (104) 



and —iff is the imaginary part of the first root of this 

 equation. 



