ON ELECTRICAL THEORIES. 117 
the currents become discontinuous, o the surface density of the electricity 
on this surface, then 
lL (u—u) + m (v—v') + 2 (w—w') + = == @, 
Remembering these equations, Y may be transformed into 
de do 
se EM ae oe 
ee yde+ [fro ds ; 
or if ¢ denote the electrostatic potential of the free electricity, we see 
wn be f(t h dgcs 
y= = I;geae 
Substituting this value of J we find 
do 
Sait 4ru, 
7°U = (1k) 
viv = (1-1) TE — Ano, 
2 
v2W = (1k) 28 — dew, 
We also see that 
dU , dV , dW ___ 1,4 
: dz dy eo dt 
In order to get the equations connecting the electromotive force with the 
variation of the electrodynamic potential, Neumann made use of Lenz’s 
law, and assumed that, since by that law the electromotive force tending 
to*increase the current in an element of circuit moving with a velocity 
w in the direction s would be of the same sign as 
—Xw, 
where X is the force along s on the element per unit of length per unit 
of current flowing through it, it was actually equal to this quantity 
multiplied by a constant c, 7.e. to 
—cXw; 
but if Ti ds be the energy of the element of current whose length is 
ds, and current strength #, 
dT 
) Ee 
ds’ 
ds 
= Bes 
so that the electromotive force per unit of length of the element 
and w 
