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VI. The Scattering of Plane Electinc Waves hy Spheres. 
By T. J. I’A. Beomwich, Sc.D., F.R.S. 
Eeceived April 13,—Eead November 23, 1916. 
Tnteoductoey Note. 
The problem which gives its title to the present paper has been handled hy various 
writers, notably by Lord Eayleigh, Sir J. J. Thomson, and Prof. Love. In most 
cases the solutions have been expressed in a Cartesian form; but it appears to me 
that a marked simplification is introduced by using spherical polar co-ordinates. The 
preliminary analysis becomes shorter, and the conclusions are easier to interpret; in 
fact, the analysis is nearly as simple as in the analogous problem of electrostatics, 
when an electric field is disturbed by the presence of a dielectric sphere. 
To obtain the requisite solutions a new general solution of the electromagnetic 
equations in Cartesian form is given in § 1, and is then transformed to the spherical 
polar form ; §§ 2, 3 contain a summary of certain analytical results required in the 
sequel. 
§ 4 contains the general solution of the problem of finding the scattered waves 
when a plane simple harmonic wave strikes a sphere; and in § 5 the solution is 
applied to the case of a small sphere. These formulae (all of § 4 and part of § 5) were 
originally worked out in 1899, hut publication was postponed in the hope of 
completing the problem of the large sphere. 
In § 6 the problem of a large sphere is considered by applying to the formulae of § 4 
a method of approximation devised hy Prof. H. M. Macdonald* for dealing with 
waves incident from a Hertzian oscillator on a conducting sphere. The formulae of 
§6 were worked out early in 1910 and were given in my University lectures at 
Cambridge in that year.f 
At the same time I succeeded in obtaining a different treatment (given in § 7 
below) which confirmed the other results, and gave an easier process for dealing with 
* ‘Phil. Trans. Eoy. Soc.,’ A, vol. 210, 1910, p. 113. Prof. Macdonald tells me that he had worked 
out (at about the same time) results in reference to the problem of § 6; but these have not been 
published. 
t An alternative solution was obtained by Prof. J. W. Nicholson at about the same time his solution 
starts from Sir J. J. Thomson’s formulae. Prof. Nicholson’s results originally differed from those of 
§6; but on revision agreement was obtained (‘Proc. Lond. Math. Soc.,’ vol. 9, 1910, p. 67; vol. 11, 1912, 
p. 277). 
VOL. ccxx.— A 576. 2 c 
[Published EebniarT 2, 1920. 
