234 
ME. J. LARMOR ON A DYNAMICAL THEORY OF 
media are non-magnetic that their rates of change along the normal (and therefore all 
their first differential coefficients) are also completely continuous. Across an interface 
the traction in the aether must be continuous, so that the tangential component of the 
aethereal force (P', Q', P') must be continuous, which is satisfied by continuity of "4^. 
The continuity of the total electric current secures itself without further condition by 
a compensating distribution of electric charge on the interface, that is by a discon¬ 
tinuity in d^f/dn. The tangential continuity of the elastic aether requires that the 
tangential component of the magnetic force (a, y) must be continuous; the normal 
continuity of the magnetic flux is assured by the continuity of (F, G, H). It might 
be argued that if the electric force (P, Q, P) were not continuous tangentially, a 
perpetual motion could arise by moving an electron along one side of the interface 
and back again along the other side. But this reasoning requires that [p, q, r) shall 
be continuous across the interface, as otherwise the circuit returning on the other side 
could not be complete; and it also requires that there shall be no magnetization, as 
otherwise the mechanical force on the electrons in an element of volume, which is 
what we are really concerned with in the perpetual motion axiom, is different from the 
sum of the electric forces on the individual electrons, by involving (a, y) iostead of 
(a, h, c). We can thus assert continuity of the tangential electric force only in the 
cases in which it is already involved in that of the tangential aethereal force ; and 
consistency is verified. The aggregate of all these interfacial electromotive conditions 
is thus continuity of the vector potential (F, G, H), and of 'F, and of the tangential 
components of the magnetic force ; they formally involve continuity of the tangential 
components of the mthereal force (P', Q', P'), and of the electric and magnetic fluxes. 
But further, in the equations from which Ampere’s circuital relation is derived above, 
it is only the normal space-variation of V' that is discontinuous ; hence continuity of 
the tangential magnetic force is involved in that of F, G, H, by virtue of the mode 
of expression of (F, G, H) in terms of the currents and the magnetism. Thus there 
are in all cases only four independent interfacial conditions to be satisfied. 
The scheme is thus far absolute, in the sense that the relations between the 
variables are independent of the special molecular constitution of the matter that is 
present. The system of equations must now be completed for material media by 
joining to it the relations which connect the conduction current in the matter with 
the electric force, and the electric polarization of the matter with the electric force, 
and the magnetic polarization of the matter with the magnetic force, in the cases in 
which these relations are definite and can be experimentally ascertained. In the 
simplest case of isotropic matter, polarizable according to a linear law, they are of 
types 
u = o-P, / = (K - l)/47rc”^ P, A = /ca. 
The expression for p leads in homogeneous isotropic media to 
KV"'F = — 47rc"p + (K — 1) [cZ/c/r (cq — hr) -f- djdy [ar — cp) -p djdz [hp — aq)} 
