188 DE. T. J. I’A. BROMWICH ON THE 
§5. Spheres Small Compared with the Wave-Length. 
The fundamental assumption is that | | is small enough to justify us in rejecting 
all but one or two terms in the power-series for S„ {ko) and E„ (/ca). 
It has been usual to assume further that | \ is correspondingly small; but 
Dr. Proudman has remarked that in the case of a dielectric sphere with a large 
value of K the second assumption need not follow from the first. It seems worth 
while therefore to simplify the formulae (47) by expanding in povmrs of |/crt|, while 
retaining the general forms for (/cja); it will be seen moreover that the resulting 
formulae form a link between the results for dielectric spheres and those for 
conductors. 
Ilemembering that (/ra) is of order and that E„(/ca) is of order (/ca)~", it is 
easy to see from (47) that in general A„ and C„ are both of order (/ca)^”"^! Thus in 
the first approximation it will be sufficient to deal only with the coefficients A] and 
Cl; and for these we need the formulae for Sj (/cia) and Sh (/ciCt). Now from (3’4) 
we have 
rj / \ 2 1 d\f sin 2 
sm 2 sin 2 /. , \ 
-cos 2 = - ( 1 “2 cot 2 ), 
and so 
Hence 
where now 
S'l ( 2 ) = sin 2 
( 2 ^— 1 -f 2 cot 2 ). 
(z) ^ 2^-l-P2COt2 _ p / 
81(2) 1—2 cot 2 ’ 
F ( 2 ) = Z^l{ 1—2 cot 2 ). 
In the second place, for the functions of /c«, from (3‘4) and (3’6) we find the first 
approximations 
Si (2) = El (2) = 1/2. 
Substituting, it will lie seen that for n — \ the first ecjuation in (47) gives 
where 
3Ai 
{Kaf ’ 
/CiOt-S^i (/fid) 
Si (/cid) 
F (/fid) -1. 
After a little reduction the last equation gives 
(51) 
A, = J (»)* 
2K-f 1 — F (/fid) 
K — 1 -t-F (/fid) 
The second equation in (47) gives a similar formula for Ci, with /x taking the 
