SCATTERING OP PLANE ELECTEIC WAVES BY SPHERES. 189 



place of K. However, in most cases, it is sufficient to write /* = 1, and then the 

 formula simplifies further and becomes 



The two formulae (5'l) and (5'2) are due to Dr. PROTJDMAN, who has pointed 

 out that they connect the results found by Lord E.AYLEIGH 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 KI may be regarded as 

 small (of the same order as KO) ; and then the approximations 



1 - (K!) cot (KIU) = -J (iqa) 2 , or F(K!) = 3, 

 may be used. This gives, in place of (5 - 1), (5-2) the simpler forms due to Lord RAYI.EIGH* 



(5-21) Al = (*<>) |^I, C 1= 0. 



On the other hand, Sir J. J. THOMSON'S case corresponds to the assumption that K is of the form 

 KI - iK 2 where K 2 is very large; then l/qaj may be regarded as large, and i as complex, with a 

 negative imaginary part. Thus approximately t cot (KI) = t, and so F (*!)! == \K l a\, which (although 

 large) is small compared with K| = j K^ | 2 /(/i)' 2 . Hence we find from (5'1) and (5 '2) the approximate 

 results, 



(5-22) A! = {{(!), d = -iM 3 



as given by Sir J. J. THOMSON, J Of course this pair of formulae follow at once from (4-71), on inserting 

 the approximations for Sj (K) and Ej (<c) given on p. 188 above. 



Dr. PROTJDMAN 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 Aj and Ci, on account of the presence in F (^a) of cot (^a), 

 which may vary very fast. It is of course supposed that the sphere is dielectric, 

 otherwise cot^a) could be replaced by t, as already stated, t 



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 



Y = + Cy = - (-f-Ci fA] COS 6) COS <f>, 



(5-3) 



Z = -c/3 = (fA,-fCi cos 6) sin ^ ; 



I KT 



* 'Scientific Papers,' vol. 4, p. 321 (106) ; see also vol. 1, p. 526. 



t Provided that the imaginary part of K^O, exceeds w in numerical value, the error in this approxima- 

 tion is less than one half per cent. 

 J ' Recent Researches,' p. 448. 



