190 DR. T. J. FA. BROMWICH ON THE 







and so (to this order) the scattered wave is zero in the direction given by 

 (5'31) = 0," cos 9 = d/Aj, 



provided that A, is numerically greater than 0,. Thus in Lord RAYLEIGH'S case, the 

 direction is given by = %ir ;* and in Sir J. J. THOMSON'S by Q = f w. 



It may be noted here that, if the sphere has a sufficiently large dielectric constant K, it may happen 

 that AI is numerically less than Ci ; and then the direction given by (5 -31) is no longer real. 



Taking K to be real (the case of a conductor having been already considered on p. 189), it is easy to see 

 that Aj < C, gives F(K I ) < 1 (on the assumption K > 1). Now the function F (z) steadily decreases 

 from 3 to as ? varies from to IT ; and a rough calculation shows that for z = |TT, F (z) is slightly less 

 than unity. Also in order to justify the approximations used for Sj (KO) and Ej (K<J), we must suppose 

 that (KO,)- ^ i^. 



Hence the possibility contemplated may occur if, say, 



(f7r)2<( Kl a)^7r2, and (i) ^ ^ 

 giving 



K > 75. 



The direction in which the scattered wave vanishes will be given by 

 (5-32) <(> = ir, cosfl = A,/C, = 2F(K,a)/{3-F (,!)}, 



the final formula being simplified by remembering that K is large. 



I am not aware that there is any experimental evidence showing traces of this phenomenon ; in fact all 

 the evidence shows that <f> = 0, = \ir is not far from the truth. Thus the circumstances in actual 

 experiments cannot have been such as to introduce the reversal of magnitude between A T and Cj. 



(ii.) Second Approximations. 



We proceed next to find second approximations, assuming that | K is not large ; 

 it will be necessary to retain the second terms in the series (3'4) and (3'6) for Sj and 

 Ej, but the first terms will suffice for S, and E.,.t It is easy to see that then terms 

 of order (/fa) 8 occur in the coefficients A,, C, and A.,, but that no other coefficients can 

 contain terms of order lower than ( K af . 



Using now the series (3 '4) for S, (z) we have 





, S', (z) = Ml 

 retaining the second terms only in each series. Thus we find, to the same order, 



Also (3'G) gives similarly 



* A closer approximation is worked out on the next page; see formulae (5-6) below. 

 t It would be possible, of course, to obtain second approximations to (5- 1) and (5-2), but a glance at 

 the formulae shows that the work is so laborious as to be almost impracticable. 



