EQUATIONS OF PEOPAGATION OF ELECTRIC WAVES. 21 
these expressions are to be formed for any time t, and then t — r/c is to be substituted 
for t in them. 
Reduction to a Single Type of Sources. 
19. We may seek to express our results in terms of sources of a single type, 
instead of using two types of sources. The method to be followed is analogous to 
that used by Green" for the Theory of Potential.* If V is a function, which is 
harmonic at all points outside a closed surface S, the value of V at an external 
point 0 is 
, r cv Gv j 
(«). 
where is the element of the normal to S drawn outwards, and r is distance from O. 
If now V' is harmonic within S, and etpial to V on the surface, this becomes 
where dv is the element of the normal to S drawn inwards. This result is obtained 
from the reciprocal theorem 
1 SW 
r dv' 
dS = 
Y' ^dS, 
Ov 
both V' and r~^ being harmonic at all points within S. Further, we know that there 
cannot be two functions satisfying the conditions satisfied by V'; the theorem that 
there is one such function is the fundamental existence-theorem of the Theory of 
Potential. The expression (40) may be interpreted in terms of sources and doublets 
on S ; the expression (41) admits of a similar interpretation in terms of sources 
only, t 
20. In adapting this process to the present theory, we begin by proving that there 
cannot l^e two sets of related vectors which 
(1) are free from singularities at all })oints within a closed surface S ; 
(2) satisfy the system of equations (12) of § 9 at all points within S ; 
(3) yield the same tangential components, for either of the two vectors, at all 
points on S ; 
(4) vanisli throughout the space within S for some value Q of t. 
If there were two such sets, their differences (a, /S, y) and [f, g, Ji) -would satisfy 
* Creen, ‘ Math. Papers,’ p. 29. 
t Lamb, ‘ Hydrodynamics,’ pp. 6G, 67. 
