798 
i\lR. J. LAK]\JOR ON A DYNAMICAL THEORY OF 
I'lie validity of the application of the LagTangian ecpiatious in the unmodified 
form to electric currents, as in the discussion in this paper, thus requires that there 
is no intrinsic cyclosis in the motions which exist in the electrodynamic held. The 
conductors must therefore all form practically incomplete circuits, in which the flow 
ujay be maintained and altered by means of what are effectively breaches in the 
continuity of the medium ; and as a further consequence, arising from such breaches 
of continuity, the mechanical forcives between the conductors will not now be wholly 
due to ordinary fluid pressure. 
In an ordinary electric circuit, the circulation of the medium is thus maintained 
around the conducting part of the circuit by electric convection or displacement 
across the open or electrolytic ])art, by means of a process in which the rotational 
elasticity of the medium is operative. We may imagine this electric convection to 
be performed mechanically, and to be the source of the energy of the current : the 
force-component corresponding to the dynamical velocity which represents the 
curi'ent will then be the electric force which does work in the convection of charged 
ions. If this convection ceased, the circulatory motion which constitutes the 
magnetic field of the current {i.e. its momentum) would be stopped by the elasticity 
of the medium ; and by altering the velocity of this convection, we have the means 
of adding to or subtracting from the circulatory motion, the change of kii:ietic energy 
so produced being derived from the electric force which resists convective' displace¬ 
ment. This mode of mechanical representation suffices to include all the phenomena 
of ordinary electric currents. On the other hand, in a molecular circuit there is no 
electric convection, but only a permanent fluid circulation through it, such as would 
Ije self-subsisting, by aid of fluid pressure only, when the core is fixed, and could not 
in any case be permanently altered, on account of the rotational elasticity. 
In the establishment of an ordinary current in an open circuit, the rotational 
elasticity of the medium acts very nearly as a constraint, on account of the great 
velocity of electric propagation ; and there is therefore at each instant only an 
insignificant amount of energy involved in it. But notwithstanding, if there are 
other open conducting circuits in the neighbourhood the action of this elasticity hi 
establishing the current will be partly directed by them and relieved by circulation 
round them. The final result for maintained currents is however irrotational motion 
through the circuits ; the kinetic energy is sufficiently represented, for slow changes, 
by the oi’dinary electrodynamic formula for linear currents ; and it is directly 
amenable to the Lagrangian analysis. If the currents move in each others’ fields, 
with external agencies to prevent their strengths from altering, these agencies must 
supply twice as much energy as is changed into mechanical work in the movement, 
in accordance with a theorem of Lord Kelvin’s. 
Conversely, assuming that the electromagnetic energy is kinetic, it would seem 
that we are required by Lease’s law to take the currents in ordinary electric circuits 
to be of the nature of velocities, in the dynamical theory ; though in the essentially 
