SCATTEEING OF PLANE ELECTKTC WAVES BY SPHERES. 
185 
the wave-train has unit amplitude ; then, in terms of the Cartesian specification, 
the incident wave is defined by* 
X == c/3 = 
the remaining components of force being zero. 
We must first express this wave in the standard forms of (2’l) and (2'2); we 
therefore introduce polar co-ordinates, and then proceed to find the radial components 
of force, which will suffice to determine the functions U, V. 
These radial components are given by 
(4‘l) X = —sin 0 cos </> ca. = -t-sin B sin ^ 
where the time-factor is now omitted. 
Now from (3‘9) we have the formula 
g.Kr cos 9 ^ £ (2n-f 1) P (cos B). 
So, differentiating with regard to B, we find that 
(4-2) 
sin = - i {271+ 1) f'-' P'n (cos B) sin B. 
(/cr) 
71 = 1 
Accordingly, on substituting (4'2) in (4’l), we find that in the incident wave 
(4-3) 
and 
y ^ _ sin 9 cos f I (2^ (^0 P'. (cos 0), 
/c" n = l {n + 1 ) 
V = + i .-.S.(,r)F.(cosfl) 
K n=l7l{n+l} 
by comparing the two formulae (2‘5) and (2‘8). 
The corresponding waves in the interior of the sphere will be given by the two 
functions 
(4-4) 
where 
and 
jj sin 0 cos </) 2'U+l „_i-d o / \ t)/ 
Ui =-J^ X —-p; (/nr) F „ (cos 0), 
/c" n=l 71 {71 -I- 1 ) 
y sill f> COS ■p _ 5 ,.-iD,S. hr) P'. (cos 0). 
n = l7l{7l+l) 
/fl = /C \/(y«K) 
and K, ju are the fundamental constants of the spherical obstacle. 
* It is assumed that in the incident wave we may take /a = 1, K = 1, Ci = c, kj 
2 D 2 
