23 
EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
same as those in the solution v, . . . h) for external space, this set also makes the 
tangential components of electric displacement the same as those in the solution for 
external space ; we take this to be the set with suffix 1, so that, on S, 
(j^n — Ap>i =: ya — hm, Gpi — = Gu — 1 
I 
E(juation (37) of § 15 now becomes 
47rC“^/o = I j dS I [w ~ iv^) — — v{)} -j- y'{n[u — uj — I {w ~ w^)} 
+ h'{l {v - 7’j) - m{u - u^)]] . . . (42). 
The corresponding equation for Uq would Ije 
47rC“hQ = dt[^ii'{m{io — ^t^j) — n{v — jq)} + v'{ii {a — '.q) — I {i.o — w^)\ 
+ iv'{l{v — i\) — in {a — ?q)j] . . . (43). 
The residts would thus ije lnterpretal)le in terms of sources of the second ty})e 
only, the radiation functions of the sources depending upon the surface values of 
u Uj, V — tq, IV — ?«q, in the same way as those in § 18 depended upon the surflice 
values of v, v, iv. We might, in a similar way, show how the general forms of the 
displacements could l3e expressed in terms of sources of the first type only. 
Tt is to he noted that this reduction of the number of types of sources has depended 
iq)on the possiljility of choosing a time, before wliich there is no distuiffiance at a]iy 
point within the given closed surface, and also that it involves an existence-theorem, 
which has not been proved. For a S])here, the existence-theorem could be proved by 
help of the formuhu in § 10. 
'Laiv of Disturbance in n Secondary Wave. 
22. As a first application of the geueral formulm we may consider the law of dis¬ 
turbance in a secondary wave. We suppose that simple harmonic plane waves, of 
the simplest type, polarised in the plane (.r, z), and propagated parallel to the axis 
of 2 , are to be resolved into secondary waves due to sources, situated on the wave 
front 2 = z. Let the primary waves be given by 
F = 0 , 
u = cos k{z — c^) , 
f=0, 
1 
G = -sin k{z — ct) , 
K 
V = 0 
H = 0, ] 
«■ = 0 A ■ • 
g = — K sin k{z — Qt) , 
? J 
