190 
DR. T. J. TA. BROMWICH ON THE 
and so (to this order) the scattered wave is zero in the direction given by 
(5‘31) = 0, cos 0 = Ci/Ai, 
provided that is numerically greater than Ci. Thus in Lord Rayleigh’s case, the 
direction is given by 0 = ^-tt ;* and in Sir J. J. Thomson’s by 0 = f tt. 
It may be noted here that, if the sphere has a sufficiently large dielectric constant K, it may happen 
that Aj is numerically less than Ci; and then the direction given by (5'31) is no longer real. 
Taking K to he real (the case of a conductor having been already considered on p. 189), it is easy to see 
that Aj < Cl gives F (/<!«) < 1 (on the assumption K > 1). Now the function F {z) steadily decreases 
from 3 to 0 as z varies from 0 to tt ; and a rough calculation shows that for z = F (z) is slightly less 
than unity. Also in order to justif}'' the approximations used for Sj (ko) and Ej (ka), we must suppose 
that (Ka)'^ ^ Jq. 
Hence the possibility contemplated may occur if, say. 
giving 
= ('^ 1 ^)^ = and (ka)^ ^ 
K > 75. 
The direction in which the scattered wave vanishes vdll be given by 
(5’32) (f> = ^TT, cos d = Aj/Cj = 2F (Kia)/{3 — F (k-ja)}, 
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 <^ = 0, d = 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^ and C^. 
(ii.) Second Approximations. 
We proceed next to find second approximations, assuming that |K1 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 2 and E^.t It is easy to see that then terms 
of order (/ca)® occur in the coefficients Aj, C\ and A.^, hut that no other coefficients can 
contain terms of order lower than (/ca)^ 
Using now the series (3'4) for Sj {z) we have 
Si (2) = I-2" (I —1L2"), S'l (2) = §2 (I - ^2^), 
retaining the second terms only in each series. Thus we find, to the same order, 
Also (3’6) gives similarly 
E,(z) = hl+i^h = 
2 2 
* 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. 
