SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
177 
Here (X, Y, Z) denotes the electric force, (a, ft, y) the magnetic force, K is the 
dielectric constant, fx is the magnetic permeability, and the axes of reference are a 
Cartesian right-handed system. The units adopted are those of the electrtnnagnetic 
system, and c is the fundamental constant generally identified with the velocity of 
radiation in free space ; the equations (E) are those derived from Ampere’s law, and 
the equations (M) are similarly derived from Faraday’s law, the two together 
constituting the circuital relations of the electromagnetic field. 
It has proved possilfie to obtain a solution of a very general type, by assuming that 
(ri) 
X = -*Q. 
dx 
Y = — 
dy 
then equations (M) yield 
( 1 - 2 ) 
■fX 
doL _ 
dz " dy 
dt 
y 
dft ■ 
dQ _ dQ 
0P ^ 
dy 0Q 0Q 
Substitute from equations (l'2) in the first equation (E) and we obtain 
fxK d^X 
(1-3) 
dft 
dx 
02 02 02 
where denotes Laplace’s operator ^ ^. 
On comparing equations (I'l) and (l'3) they will be seen to be consistent provided 
that 
(1-4) 
^xK 0"P 
c" dft 
and that 
(1-5) 
mK 02Q 
r\ 3Q 
dx dy 
aQ 
dz 
A^Q. 
Thus Q must satisfy the fundamental wave-equation, which is satisfied by any 
component of the electric or magnetic forces (X, Y, Z) or (a, ft, y). 
For our purpose it is more convenient to express the above solutions in terms of 
spherical polar co-ordinates r, 0, (ft; these are supposed to form a right-handed^system, 
when taken in this order, so as to avoid changes of sign in introducing the new 
co-ordinates. We write here"^ (Rj, 11-2, fh© components of electric force in the 
directions of r, 0, ^ respectively; and (Hi, Ha, Hg) for the components of magnetic 
force. 
Equations (I'l) then become 
(rii) 
^ _ 0P ^ 
“ 0r 
R 2 
1 ^ 
r 00 
1 0P. 
r sin 0 0^ 
* This is done to avoid confusion with the Cartesian components used in equations (E) and (M) ; but 
^in the subsequent sections we shall use (X, Y, Z) and (a, ft, y) for the spherical polar components 
here denoted by (Ri, R2, R3) and (Hi, Ho, H3) respectively. 
2 c 2 
