See 
ON ELECTRICAL THEORIES. 119 
so that if the conductor has an infinitely small resistance, the equation 
becomes id 
24—A2.%% 
b= A*k Te 
This represents a wave motion, the velocity of propagation of which is 
1/AVk. If k, as in Neumann’s theory, be equal to unity, then the 
velocity of propagation is 1/A, and from the value of A, found from 
experiments on the force between circuits conveying currents, this is 
nearly equal to the velocity of propagation of light. Thus, according to 
Neumann’s theory, in a perfect conductor an electrostatic disturbance is 
propagated with the velocity of light. In an insulator ¢ satisfies the 
equation 
V?o=0; 
and this represents a motion propagated with an infinite velocity, and 
thus, according to this theory, an electrostatic disturbance is propagated 
with an infinite velocity in a perfect non-conductor. In an imperfectly 
conducting substance the velocity of propagation of a wave motion would 
depend upon the length of the wave. ; 
Let us now go on to consider, what, according to this theory, are the 
forces acting on an element of circuit conveying acurrent. Let us suppose 
that the element ds forms an element of a circuit through which a current 
tis flowing ; then the energy of the circuit will be 
dx dy dz | 
2 Se ae — 
A |: {usev awe | as, 
In order to find the force parallel to z, let us suppose that each element 
of the circuit receives an arbitrary displacement 2, parallel to the axis of 
z; then the alteration in the energy will be 
5 f dU da dV dy aes : = d da 
atl bm Engh aa" aide dz ds + A? 1. U a ds. 
Integrating the second term by parts, we see that it may be written 
5 dU dz , dU dy , dWdz 
2 SNe 4 —  _ —_ 
[A2.Udce] —A | oe fois ia ds se ee at dw ds. 
Substituting this value for the second term, we see that the alteration in 
the energy, 
dy (dV _dU\ dzfdU dW). 
Se A2 2 ef OL 7 Ne ee ; 
~ eahaaniae ladle | * lds \de dy) ds\dz dz ) pe ip 
hence we see by the Conservation of Energy that there is a force on each 
element of current parallel to the axis of x, equal to 
fy dV_dU)_ dz BCL yr 
ds\ du dy ds\dz dx 
and by symmetry forces parallel to y and z equal respectively to 
pees (a PN -2(2-2) Vas, 
ds\ dy dz ds\du dy 
SdujdU0 dw —n v) bas 
"Vas dz du ds “dy dz 
