A PLANE ELECTROMAGNETIC WAVE BY A PERFECTLY CONDUCTING SPHERE. 281 
Using spherical polar co-ordinates r, 0, 0, the incident plane wave train is taken to 
contain the time factor e lKct , to travel along the negative direction of the axis 0 = 0, 
to be polarized in the plane 0 = |-7r, and to have the electric force of unit amplitude. 
The components of electric and magnetic force, in the disturbance produced by the 
sphere, along the directions of r, 0, 0 increasing, are denoted respectively by X, Y, Z 
and a, /3, y. Then at a distance from the sphere, which is large compared with a 
wave-length of the incident train, we have, from the paper quoted, 
where 
X = col = 0, 
8M 3N 
Y = Cy = 
0 sin 000' f 
7 _ _ n _ 0M 0N 
~ sin 000 00 ’ 
( 1 ) 
M 
N 
= COS 0 ■ 
• > ^ e 
sin 0 - 
Take now 
——' sin e £ (-1 r- ■■ 0- Yi i It 4A P'. (cos 9), 
kv « = i n (n+1) hi n (Ka) 
- iK (r-ct) - 2n + 1 S„ (kO) -p/ , .v 
-sm 0 X (-1)” +] - yv " V ■ \ P , t (cos 0). 
kv n = i n (n+l) k, n (i<a) 
( 2 ) 
_t« (r—ct) 
Y = cos 0 -(Yj+t’Yj,), 
Z = sin 0 
KV 
— ik (r—ct) 
y 
(3) 
(Z 1 + zZ 2 ), 
/cr J 
where Y ls Y 2 , Z l5 Z 2 are real. Then the time-means of the squares of the real parts 
of Y and Z are respectively 
j cos 2 0 
2 
(Y^+Y./), 
] sin 2 0 
2 w 
(Z.’ + Z/) 
These give a measure of the energy of the disturbance. 
The functions Y l5 Y 2 , Z 1? Z 2 involve only k a and 0. It is the object ot the present 
paper to evaluate them, together with Yj 2 + Y/ and Zy + Z ./ for certain values of <a, 
and all values of 0 from 0 to -rr. 
Writing cos 0 = n, we have 
wr {sin 0P / „(^)} = ra(tt+l)P n M—/*P'»M» 
using which we obtain 
Yj = A+mA'-C', Y 2 = B+mB'-D',' 
Z, = A' + /iA-C, Z 2 = B' + mB-D, 
2 r 2 
( 5 ) 
