EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
19 
17. It may be verified without difficulty by using Ktrchhoff’s method that the 
integrals written down in § 16, when taken over a closed surface S, not containing 
the point {x, y, z), but containing all the singularities of the functions v, v, w,f, g, A, 
represent the values of 47r«, ... at the external point {x, ?/, z), provided the normal 
(/, m, n) is drawn towards the exterior of S. It may also be verified, in the same way, 
that, if the surface S contains the })oint [x, ?/, z) and all the singularities of tlie 
functions, the integrals in question vanish identically. 
A particular case, which leads to a verification of the formulre, is afforded l:)y 
taking the surface S to he a sphere, of radius Qt, having its centre at the point [x, y, z). 
For this we have 
X — x' = Ir, y — y' = mr, z — z' = nr, r = eg 
and the values of the quantities v, r, , at points on S, are the initial values, 
Uq, Vq, . . . , of these quantities. Now the terms of 47ru that contain u, r, u\ 
explicitly are 
r 2||dS[Wo - /(/'Q + 
which = r " |1 ^ j dS (x — x') (Iuq + + nWo) 
cISkq — / 
' n,<lr + 111 (,r - a-') (■£'; + ),/r, 
where the volume integrations extend through the volume within S, and the last 
volume integral vanishes identically. Again, the terms that contain F, G, H 
explicitly are 
wide] I 
ugh. 
Further, the terms tliat contain F, G, II explicitly can he written 
j” c/S {mJiQ — 
by observing that F = c‘^V~F = — hf, . . . 
