816 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
together with internal stress as yet undetermined between contiguous parts of the 
conductors, constitute the total electromagnetic forcive : it would not be justifiable to 
calculate the circumstances of internal mechanical equilibrium from the Ampereau 
forcive alone, unless the circuits are rigid. For example, if we suppose that the 
circuits are perfectly flexible, we may calculate the tension in each, in the manner of 
Lagrange, by introducing into the equation of variation the condition of inexten¬ 
sibility. We arrive at a tension ffMfVZs', where i is the current at the place 
considered ; whereas the tension as calculated from Ampere’s formula for the forcive 
would in fact be constant, the forcive on each element of the conductor being wholly 
at right angles to it. 
The general case wdien the currents are not linear is also amenable to simple 
analysis. The energy associated with any linear element ids is idsllslids'; which is 
equal to ids multiplied by the component of the vector-potential of the currents in 
the direction of ds, when the conduction and convection currents move round 
complete circuits. Thus, changing our notation, the energy associated with a current 
{u, V, lu) in an element of volume dr is (Fii -j- Gv -j- Hiy) dr. In this expression 
(F, G, H) is the vector-potential of the currents; if there is also magnetism in the 
field, there Vvull be a part of this vector-potential due to it, which may be calculated 
from the equivalent Amperean currents. Thus for a single Amperean circuit, 
F — dx, which by Stokes’ theorem = iKixdjdz — vd/dij) dS, where (X, /x, v) 
is the direction-vector of the element of area dS ; hence the magnetic part of the 
vector-potential is {Bdjdz — Gdjdy, Gdldx — Kdldz, kdldy — V>djdx) which 
agrees with the assumption in § 110. It will be observed that in the vector-potential 
of the field, as thus introduced, there is no indeterminateness ; it is defined by the 
expression for the energy, as above. 
We may complete this mode of expression of the energy by including the energy of 
the magnetism in the system due to the field in which it is situated. For a single 
Amperean atomic circuit it is f{(Fda; -}- Gdy fi- 11^2:), which is by Stokes’ theorem 
■i j{\ (c/H / c/y — dG j dz) c/S ; thus the energy of the magnets is 
|(Aa + Byd -f- Gy) dr, where (a, /3, y) is the magnetic force due to the external 
field as usually defined; this follows from the formulae for (F, G, H) already obtained. 
There is also the intrinsic energy of the magnets due to their own field; by the well- 
known argument derived from the work done in their gradual aggregation, the 
co-ordinated part of this is ^ KAa^ + fi- Cyo) where /Sq, yo) is the force of 
their own field. These terms will add on without modification to the other part ol 
the electrokinetic energy for the purpose of forming dynamical equations, provided 
we assume as above that the magnetic motions are not of a purely cyclic character. 
This sketch will give an idea of how magnetism enters in a dynamical theory which 
starts from the single concept of electrons in movement. 
The energy being thus definitely localized, and all the functions precisely defined, 
we derive in the LaoTano-lan manner the electric force 
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