ON ELECTRICAL THEORIES. 11E 
_ expression for the induction due to the motion of the primary circuit, or 
variation in the strength of the current passing through it. 
Frohlich ' urges against Clausius’ law that since, according to it, an 
electric current in motion exerts an electromotive force on a moving 
electrified particle, even though the particle is moving at the same rate- 
as the circuit, every current on the earth’s surface ought to exert an 
electromotive force on an electrified particle relatively at rest, since each 
is moving with the velocity of the earth. This force is one that can 
be derived from a potential, so that the integral of the force taken round 
a closed curve would vanish, and thus, even if this result were true, two 
circuits would not induce currents in each other if they were relatively 
at rest. Budde? points out, however, that the moving circuit would exert 
an electromotive force at each point of itself, and thus cause a separation 
of the electricity in the circuit, so that it would get coated with a distri- 
bution of electricity, tbe electrostatic action of which would balance that 
due to the action due to its motion on a point relatively at rest. The 
velocities which enter into Clausius’ formule are velocities relative to the 
ether, so that if the ether moves with the earth, an electric current will, 
according even to this theory, exert no electromotive force on a point 
relatively at rest, and there will be no electrification on the surface of 
the circuit. The velocity ec which occurs in all these theories is a velocity 
comparable with the velocity of light. 
General Considerations on these Theories.* 
We shall now go on to discuss a general way of treating theories of 
_ the kind we have been considering. Perhaps the best way of doing this 
_ is to consider not the forces between the electrified bodies, but the energy 
possessed by them. If the energy depends on the electrification there- 
_ will be forces between two electrified bodies. Now the potential energy 
depends on the electrification, and this dependence produces the ordinary 
electrostatic forces between two electrified bodies at rest. If, however, 
the kinetic energy as well as the potential depends on the electrification, 
then the forces between two electrified bodies in motion will be different 
from the forces between the same bodies at rest. An easy way of seeing 
this is by means of Lagrange’s equations. 
If T be the kinetic energy, and w a co-ordinate of any kind, then we 
have, by Lagrange’s equations, 
Aly 
d av _ dv = external force of type z. 
di dé da 
Hence if we have any term T’ in the expression for the kinetic energy,. 
we may, if we like, regard it as producing a force equal to 
4 at at 
dt dz da” 
TS 
A simple illustration of this is afforded by the centrifugal force. In 
1 Frohlich, Wied. Ann., ix. p. 277, 1880. 
2 Wied. Ann., x. p. 553, 1880. 
3 See Clausius ‘On the Employment of the Electrodynamic Potential for the 
Determination of the Ponderomotive and Electromotive Forces,’ Phil. Mag., 1880, v. 
10, p. 255. 
