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VI. The Scattering of Plane Electric Waves by Spheres. 

 By T. J. I' A. BROMWICH, Sc.D., F.R.S. 



Eeceived April 13, Read November 23, 1916. 



INTRODUCTORY NOTE. 



THE problem which gives its title to the present paper has been handled by various 

 writers, notably by Lord RAYLEIGH, 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, but 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 formulse of 4 

 a method of approximation devised by Prof. H. M. MACDONALD* for dealing with 

 waves incident from a Hertzian oscillator on a conducting sphere. The formula of 

 6 were worked out early in 1910 and were given in my University lectures at 

 Cambridge in that year.t 



At the same time I succeeded in obtaining a different treatment (given in 7 

 below) which confirmed tlie other results, and gave an easier process for dealing with 



* 'Phil. Trans. Roy. 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 formulse. Prof. NICHOLSON'S results originally differed from those of 

 6 ; but on revision agreement was obtained (' Proc. Lend. Math. Soc.,' vol. 9, 1910, p. 67 ; vol. 11, 1912, 

 p. 277). 



TOL. CCXX. A 576. 2 C [Published February 2, 1920. 



