SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
179 
Substituting from (l‘6) in (i'll) and (l'2l) we obtain the final expression for the 
field in terras of U :— 
R, = 
(1-8) R2 = 
R, 
a^u 
07-=^ C 
1 a^u 
r Bo B?' 
1 a^u 
juK a^u 
2 
P B<p Br 
Bt^ 
cHi 
cR, 
CH3 
= 0 
= + 
I a ^ K a u 'j 
pB(p\c Bt 
i_a/Kau' 
7'B0\c Bt 
p = r sin 6, 
In like manner we obtain another set of solutions by making an assumption similar to 
(I'l) for the components of magnetic force (a, y). This gives the field :—- 
(1-9) 
1 B (p av \ 
p B<p\c Bt J 
R, — + 
r B0\c Bt ) 
cH, = 
Br^ 
pK a^v 
c" Bt^ 
cH3 = 
1 a^v 
r BO Br 
a^v 
00 Br 
> 
p = r sin 0, 
where V is a second solution of equation (17). 
It can be proved* that (l'8) gives the most general field in which the radial 
magnetic force (Hi) is zero, while (l'9) gives the most general field in which the 
radial electric force (RJ is zero. It can also be shown that the field is uniquely 
.determined by the value of Ri and Hj; and accordingly the most general solution 
can be obtained by the superposition of (l‘8) and (l‘9). 
§ 2. Further Specialization of the Solution of § 1. 
If we superpose the fields (P8), (l‘9), and now utilize (X, Y, Z), (a, /3, y) to denote 
the spherical polar components of the field, we have the general solutiont :— 
( 21 ) 
X 
Y 
Z 
a^u 
ar^ 
pK a^u 
Bt^ 
= 1 _ i A 
r Bo Br p 00 \c 0^ ) 
^ 1 a^U I B UBY 
„ ^-J. 7\/v» /v» \ 
p = r sin 0, 
* See a paper in the ‘Philosophical Magazine,’ July, 1919 (6th ser., vol. 38), p. 143. 
t Originally worked out in 1899, and first published as a question in Part II. of the ‘ Mathematical 
Tripos,’ 1910. 
