Elect rh: Waves round a Large Sphere. 169 



in E ft and <f> n for the Bessel functions, it is seen that the poles 



are given by the solution of the equation in m, 



3K./3: = -'<• (92) 



and since, whether m be real or not *, 



K„ = —I K (2z bmh t) cosh 2mtdt, 

 "Vo 



we note that R„ is an even function, so that m= — v i> 

 also a pole. The poles thus occur in pairs. Moreover, an 

 inspection of (80) shows at once that it cannot be satisfied 

 for real values of m when z is real. Thus there are no real 

 poles. Now when ra, even though complex, is not nearly equal 

 in modulus to z, although less, 



R. = */(**- m«)i 



and it is found to be impossible to satisfy (92) within the 

 limits of validity of this formula. The poles therefore 

 correspond to values of m of order z at least. 



We may assume, at once, the justification appearing in the 

 result, that the poles contributing mainly to the sum are 

 those whose imaginary parts are least. Moreover, it is fairly 

 evident from the above reasoning that the least order the 

 imaginary part can have is that of zi, since there are no poles 

 in the first region of expansions of the Bessel functions. In 

 the section following, the first pole is determined and found 

 to be of the form 



m = :-^/3 , (93) 



where /3 is a numerical quantity approximately equal to 1/3, 

 and its contribution to the sum is of supreme importance. 

 We shall also, for the present, assume that the imaginary 

 part of the poles is negative. If a v be the residue of the 

 function 



zlK m (z) & .*«.(*) = iiR n (l + * 2i *») 



at the pole v, then the corresponding terms in its development 

 by Cauchy's theorem, including the pole — v, become 



-(—-) = 



\m — v ))t -\-vj 

 Vide Phil. Mag. Feb. 1910 for the case of m real. The proof there 



,va v 



given can obviously be extended. 



