SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
191 
On substituting these results ( 47 ) becomes 
2 _FU£!(i _ A (1 + _ A, A 
K 5 
and 
P 
10/ p 
where, for brevity, we have written 
tea = p, {k-plY = Kjol 
On reducing these equations, the results are 
A — 
■^1 — sP 
(5-4) 
K-1 
1 + i 
K-2 
and 
^K+2/ r nK+2 
Cl = 4V(K-l)y. 
p‘ 
To determine Ag, the series for Sg {z) and Eg {z) will be required to the first terms 
only ; these are 
and on substituting in ( 47 ), we find 
(5-5) 
A ^ pVK-1 
" 15\2K + 3 
The field of the scattered waves is then given by 
Y — pcy = -(fCi—|Ai cos 0+f Ag cos 20) cos 96, 
I kT 
(5-6) ■; 
1 Z = -c/3 = - —^(|A,-|CiCos e-fAjCos 9 ) sin </>. 
I KV ‘ 
The field (5‘6) is accordingly zero {to the same degree of approximation) hi the 
direction given hy 
0 = 0, cos 0 = (Cl—fAg)/Ai = 2K + 3 ^ (^a)^- 
This conclusion is apparently new ; but it confirms an approximate result due to 
Lord Rayleigh,* according to which the scattered wave is zero in the direction 
given by 
0 = 0, cos 0 = xV(K—l) [xaY, 
when (K—l) is treated as small. But on the other hand, our result contradicts 
* ‘Scientific Papers,’ vol. 1, p. 531, formula (61). 
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VOL. eeXX.—A. 
