SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
183 
Similarly, we write 
(3-5) = 
= c„(^)-.s„(4 
where 
(3-6) C,(.) = .-(-i|)*(52p) ' . 
_ 1 ■ 3 ... (2n-l) \ _2^_1 
2" 2 (l-2n) ^ 2.4(l-2w) (3-2n) 
In terms of the K„ function (the modified Bessel function used by Macdonald) 
we have the relation 
(3-51) E.(z) = a /(^) 
Thus 
E„(2) = V — iU 
in terms of the notation used by Macdonald in the paper last quoted, and in 
Lamb’s notation 
(37) 
E,(2) = {'^.(2)-|^/..(z)} = 2”‘/,(2). 
In consequence of the equation (2'6) we see that both S„( 2 ) and E„( 2 ) are solutions 
of the differential equation 
(3-8) 
cU 
+ 1 - 
n (n +1) 
0 , 
The functions S„ ( 2 ), C„ ( 2 ) and |E„( 2 )| have been tabulated from 2 = 1 to 10, and 
for values of n ranging from 0 to 22, by Mr. Doodson,"^ and these tables have formed 
the basis of the numerical calculations mentioned, on p. 176 above, t 
It will be convenient to collect here the simple relations amongst the functions 
S„_i, S„, S„^i, which correspond to the known results for Bessel functions, or to those 
given by Lamb for the equivalent function \Jr„ ( 2 ). 
Differmce Relations fw the Functions S,v, E,,. 
From (3'4) we see that 
(3-81) 
and by using (3‘5) we see that the same relation holds for E,j( 2 ). 
Again, it will be found that 
(4- + = (2?l + ») 
\dz z) 
* ‘British Association Report,’ 1914. 
t Proudman, Doodson and Kennedy, ‘Phil. Trans. Roy. Soc.,’ A, vol. 217, 1917, p. 279. 
VOL. CCXX.-A. 2 D 
