118 REPORT—1885. 
v. Helmholtz has shown that it follows from the principle of the Conser- 
vation of Energy that if the energy in the elements da, dy, dz, traversed 
by currents uw, v, w, be 
A? (Uu+Vv+ Ww) da dy dz, 
then the components of the electromotive force parallel to the axes 2, y, z 
respectively, due to the variation in the electrodynamic potential, will be 
ajgam lav Saw. 
dt’ dt dt ? 
the free electricity produces an electromotive force whose components are- 
__ dd do dd 
dz’ dy ; dz? 
so that the total electromotive force parallel to 2, y, z 
dd dU 
— ee 
dx dt 
Now if o be the specific resistance of the conductor, ow equals the elec- 
tromotive force parallel to the axis of a, so that 
—_% _ 42 WU, 
hae alae = ae” 
so that by the preceding equations 
vi Orn aheey > dy eVindd=) dU 
read ey ae a 
with similar equations for V, W. The quantities U, V, W and their 
first differential coefficients with respect to w, y, z are continuous, and 
these equations enable us to find them if we know the value of 4,. 
the potential of the free electricity. Helmholtz shows that the whole: 
energy in the field due to the currents may be written 
A? dU dV\?, (dV_dW?, dW _ dU? dp 4 
al ee Nee ee LAS eet yy (ae da dy d 
alli dy =) ee = Hs ad = (Zi) ee 
so that if & be negative, this expression may become negative, and in 
that case the equilibrium would be unstable; hence we conclude that: 
only those theories are tenable for which k is positive. 
The equations written above are those which hold in a conductor, 
in an insulator the equations are 
rape? 
dedt 
do 
2V = is) 
v?V=(1 Sagat 
d*¢ 
2Ww =(1—k) — 
pa fe dz dt 
ep — 0. 
v. Helmholtz shows that in the conductor the electrostatic potential & 
satisfies the equation 
2 7 dp _ no, 0p 
‘ {e+ eth ance 
