Electric Waves round a Large Sphere. 763 



But for <j) n we must write 



*»=6 + H; ( *- m)+ siE;— 2 + --- (185) 



where R« _1 and its derivates are taken at z = m, the 

 derivates being found from 



«-wK)+ r ©o-> (*/+•••} <«' 



Thus 



6.3*w*r(i) 





r 2 (!)^ 



the latter having an order ?n *. The higher derivates of 

 R w -1 thus formed rise successively in order of smallness 

 byra - ^. They are multiplied by successive powers of z—m 

 or lz*j3 in (185), so that, effectively, 



*'-f- fe ^{ 1+ «^ + ^ + -'}' (187) 



where the a's are real and numerical. The series in brackets 

 is of course not convergent, being asymptotic in the usual 

 sense. By restricting the series to one or two terms, we 

 obtain the order of magnitude of the imaginary part of <j> ny 

 which is + l/3S, where 8 is positive, real, and numerical, and 

 also not large. Thus the imaginary part of -<f> n is of no order 

 in z*, and may be ignored in comparison with that of cf> nr . 



If <£ n were needed explicitly, the expansions of the Bessel 

 functions in terms of Airy's integral and its associates would 

 be used, as in the investigation of the transitional region. 

 But this determination of <j) n may be omitted, as it leads to 

 no points of special interest. 



Finally, therefore, (j> n —(j>nr has an imaginary part which 

 is given effectively by + tz^fi cos -1 a/r from (184), and a real 

 part which contributes only to the phase of the disturbance 

 in the shadow. A discussion of this phase, whose expression 

 is cumbrous, does not serve any useful purpose, and may be 

 omitted also. 



It follows that the real part of — i(<j> n — $>nr) is z*fi cos -1 a/r, 

 and this introduces a factor exp. z*/3 cos -1 a/r into the ampli- 

 tude of the disturbance at a distance r. This factor is unity, 

 as it should be, when r — a. The amplitude of 7 at a point 



