Electric Waves round a Large Sphere, 761 



Let m-=.y be a typical singularity of the function 



iL(U n R nr )\l + e 2l ^y(<Pn--Kr) . . . (178) 



in the plane of the variable m, now complex. All such 

 singularities must be simple poles, for (j>n — <Pnr has no 

 infinities, since R„ and R nr , by their expressions as definite 

 integrals, have no zeros (for finite values of m) y and in other 

 respects the function does not differ from its old value for 

 points on the surface of the sphere. This value, as we 

 showed, only led to simple poles. 



Since the function is even, m= —v is also a pole, and if a v 

 is the residue of the function at m = v, the corresponding part 

 of its development by Cauchy's theorem is 



/ 1 1 \ 2va v 



av I 1 - = "I r 



\m — v m + v/ mr — v z 



Thus the value of <yp is 



yp = G(0) £ m(R n R wr )*(l + e 2l X») sin m<f>e<*n-<l> nr ) 



= -iG(0)X v 2va v X .... (179) 



as in (95), with the new meaning of a„, m taking half 

 integral values in the last summation, which may be effected 

 as before. 



The first pole is again given by 



and will be found to be the only important one. As in (97), 

 we find ultimately 



7 p= - 1^0 (27Tsmey^ v v^a v e- lve+ ^ . (180) 



at a point whose orientation is 6. 



Now for points on the surface, the value of a v is z*/2i/3 as 

 in (98), where ft corresponds to v, so that v r = z — izift r defines 

 the rth pole, and if the order r of the singularity is unity, 

 ^ = •696, as in (107). The residues are not of an oscillatory 

 type in this case. But in the present problem, a is the 

 residue of a function whose exponent oscillates, so that a v 

 can oscillate in passing from one singularity to another. 

 If, therefore, this exponent should possess a zero point, or a 

 minimum point as defined in the section dealing with the 

 transitional region^ the sum of the series (180) would not be 



