186 
DR. T. J. I’A. BROMWICH ON THE 
Similarly the scattered waves will be given by the two functions 
(4'5) and 
I 
The boundary conditions are given by the continuity of the tangential components 
of electric and magnetic force at the sphere r = a. 
It is evident from the form of equations (2’l), (2’2) that these conditions will he 
satisfied if we take 
r 
I 
(4’6) and 
Thus we find that A„ and C„ (the coefficients in the scattered waves) are given by 
f {S. (,«) + A,E. (,a)| ^ 14^ = S'. (.«) + A.E', M 
(4’7) 'I and 
I {s. («) + C.E. (,«)} A. 14^) = S', (.a) + C.E'. (.a). 
The special case of a pei'fectly conducting sphere is given by making the tangential 
electric force zero at the sphere r = a\ and this condition is satisfied if 
0 = V + Vo. 
S„(/ca) + C„E„(/fa) = 0, 
which may be regarded as limiting forms of (47), when | K | -> co and ^ 0. 
The formulae (4’5) and (471) lead at once to those quoted by Dr. J. Proudman* 
in calculating the pressure of radiation due to a plane wave incident on a small 
conducting sphere. 
In all the applications with which we shall be concerned at present the point at 
which the disturbance is to be calculated will be at a distance large compared with 
(4-61) 
^ ^ TU ^ aUo 
0 ', 
r 
Thus we find the simpler formulae 
(471) S'„ (/ca) + A„E4 (act) == 0 
0Ui 
au. 
dr 
dr 
dr 
0V, 
av , 
aVo 
dr 
dr 
ar ’ 
KUi - U + Uo 
- V + V, 
TT ♦ Sm 0 cos 0 277. A 1 71 — 1 A XT' / \ TI)^ / 
Uo =-^^ 2 ■ , ' z" 'A„E„ (vr) F„ (cos 0 ) 
K n = \ n\n+l) 
_ sin B sin 0 ^ 2n+1 „_i 
0 - + ' 2'" ^ 2 {kv) P4 (cos 0). 
n = in{n+l) 
* ‘ Monthly Notices of the Royal Astronomical Society,’ vol. 73, 1913, p. 53-5. 
