Conway—Llectro-magnetic Mass. 5d 
which is zero; and we get 
— w (8r)1 VX"? 4+ VY? 4 Z?— X'? — YY’? — Z?) 
— w(87r)1V (a? + B? - a” — Bl”) + (47) 1(B'X’ — a’ Y’) 
— (4) (B"X" = a" Y") 
or if 7”, 7” denote the values just outside and just inside the surface of the 
expression which we called before the electro-magnetic energy, and WV’, WV”, 
the values of the Poynting vector resolved along the normal, the expression 
for the activity becomes — {j/w(Z" — 7”) dS - {jf (N’ — N”) ds. 
Suppose, in order to fix our ideas, that the moving surface is closed and 
contained with a fixed surface, the element of which we denote by dz, there 
being no electricity inside = except on S. We have then to prove that the 
above expression is identical with 
(J Wds + dldt {ff dr’ + ajdt {{j P"dr", 
where the second integral refers to the space inside , and outside S, and the 
third integral to the space within S. 
We have in fact 
djdt{{\T'dr' = {{{oZ"/ot dz’ — {fwT"dS = || Nd> — || N’dS — |jwT"ds, 
an 
djdt\\{ Pdr" = {| N"dS + \fwT"ds, 
so that the result follows at once. 
V.—TuHeE RETARDED POTENTIAL. 
The electric force can be expressed in form 
V(X, Y, Z) = — (0/ex, d/oy, d/oz) p - o/0t(F, G, H), 
and the magnetic force is given by 
(a, B, y) = Curl (F, G, 1), 
where w is the scalar potential, and (f, G, H) is the vector potential. In the 
case of a single electron the solutions are given by the retarded potential.* 
For slow motions it will be found convenient to express the retarded potential 
in terms of the distance, not at the retarded time, but at the actual time; 
and in doing this, we can at the same time expand in inverse powers of JV. 
* Cf, Proc. London Mathematical Soc., ser. 2., vol. i. 
