18 !) 
SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
place of K. However, in most cases, it is sufficient to write — 1 , and then the 
formula simplifies further and becomes 
The two formulse (5‘l) and (5'2) are due to Dr. Proudman, who has pointed 
out that they connect the results found by Lord Rayleigh for the case of dielectric 
spheres, and by Sir J. J. Thomson for conducting spheres. 
To deal with the case of dielectric spheres we do not regard K as large, so that may be regarded as 
small (of the same order as Ka ); and then the approximations 
1 - (ki«) cot (K'la) = ^-(Kitt)2, or F (ki«) = 3, 
may be used. This gives, in place of (5’ 1), (5-2) the simpler forms due to Lord R.A.YLEIGH* 
(5-21) Ai = f(Ka)s|^, Ci = 0. 
On the other hand, Sir J. J. Thomson’s case corresponds to the assumption that K is of the form 
Ki - iKo where K 2 is very large; then Liuj may be regarded as large, and K-^a as complex, with a 
negative imaginary part. Thus approximately! cot (/cja) = t, and so |F(Kpf)| = |K^a |, which (although 
large) is small compared with iK] = |/v'^ap/(Krt)-. Hence we find from (5'1) and (5’2) the approximate 
results, 
(5-22) Al = t('<a)^ Cl = -L(Ka)3 
as given by Sir J. J. Thomson.! Of course this pair of formulae follow at once from (4-71), on inserting 
the approximations for Si {kli) and Ei (ka) given on p. 188 above. 
Dr. Proudman makes the further remark that, under the conditions assumed in 
(5'l) and (5'2), variations in the wave-length may produce very considerable changes 
in the magnitudes of Ai and Ci, on account of the presence in F {k^ci) of cot (/cja), 
which may vary very fast. It is of course supposed that the sphere is dielectric, 
otherwise cot(AriG) could be replaced by i, as already stated.! 
It is worth while to note the simple formulae for the scattered wave, derived from 
( 4 ’9); these give, to the present order of approximation 
(5-3) 
i Y =i -\-cy — -(fCj—-|Ai cos 0) cos (i>, 
I kV " “ 
Z = —c/3 — -(dAi—dCi cos 0 ) sin (p ; 
/cr “ 
* ‘Scientific Papers,’ vol. 4, p. 321 (106); see also vol. 1, p. 526. 
t Provided that the imaginary part of kiu exceeds tt in numerical value, the error in this approxima¬ 
tion is less than one half per cent. 
I ‘ Recent Reseai-ches,’ p. 448. 
