SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
187 
the wave-length. Then we can simplify the general formulae by observing that (3'5) 
may be replaced by the approximation 
E„ {kv) = 
if I/kv, {ifKrf, &c., are neglected. Further, in the final formulae for the forces, 
Uo and Vo occur only in the two combinations 
M = 1 _ I = 'aYx , 
•y 07’ T V O)' V 
where terms of the relative order 1/ kv have been rejected. 
On substituting from (4'5) we find, to the same degree of accuracy. 
r M = + Z ^ = - - Uo = sin d cos <p 
I r dr r kt n = \ 
(4-8) <; 
and 
N = - 
1 8Un 
r dr 
Z = + - Vo = sin d sin (p 
r dr r 
— iKr <x> 
2 (-1)'‘tw^^A„F„(cos0) 
n{n+l) 
_tKr CO 
^ v / 1 271-f-l_ 
.r n{n+l) 
C„F„ (cos 0). 
Then, substituting in the general formulae (2T) and (2‘2), we find that (to our 
present order of approximation) the radial components of force are zero, and that the 
transverse components are given by 
0M 1 0N 
-^-^ = +Cy 
do sm 0 d(p 
sm 0 d(f) do 
Accordingly the electric and magnetic forces in the scattered waves are at right 
angles to each other and to the radius, a^id their magnitudes are related in the same 
maimer as in a plane ivave. 
This conclusion might very well have been anticipated ; and for the case of smcdl 
obstacles of any shape (with constants K, fx differing but little from unity) the 
conclusion is contained in a paper by Lord Rayleigh."^ But I cannot find that it has 
been noticed for the case of spheres of any size, and of any electrical and magnetic 
constants. 
This may serve to indicate one advantage of the formulse m spherical polars over 
those in Cartesian co-ordinates. 
The formulse (4-8), (4-9), with the values of A„, C„ given by (471), were those 
used by Messrs. Proudman, Doodson, and Kennedy in their numerical calculations 
quoted in the introduction to this paper. 
* ‘ Scientific Papers,’ vol. 1, pp. 522-536. For a small perfectly conducting sphere the same conclusion 
is given by Sir J. J, Thomson, ‘ Recent Researches,’ p. 448. 
(4-9) 
Y = 
Z = 
