Maxwell’s and Lorentz’s equations take, in
the case of pure ether, a form remarkably analogous to the ones for the
equations of elasticity. What is the real significance of this analogy?
The electric vector E is satisfied in
ether by the equations
and likewise for H. This is the immediate
consequence of the fundamental equations (I) through (IV).
On the other hand, the components of
displacement, x, h, z, assumed to be small with respect to a point in the
elastic body, A and B being constants, and m the density;
Electromagnetic theory shows, as we
know, that E is identical to Fresnel’s vector, H to Neuman’s vector (parallel
to the plane of polarization). This identity with systems (14) and (15) leads
to an elastic
theory of light. To do that, we have to
admit either the incompressibility of ether, that is to say the condition
or the condition A + B = 0. In both cases,
the identification is immediate. These two ways of explaining the
non-existence of longitudinal waves have been admitted. In each of these
hypotheses we could again choose between Fresnel’s theory which leads to
identifying the speed
of ether with E, or Neumann’s theory which
replaces E with H.
What are the general conditions,
necessary and sufficient, such that a physical phenomenon, characterized by a
vector, will follow the laws expressed in (15)? I say that they are the
1. The phenomenon is reversible.
2. x, h, z satisfy a system of three
partial differential equations which are of second order at most, and at least
are linear to a first approximation.
3. The medium is isotropic and
Indeed, in considering reversibility,
the equations don’t have first derivatives with respect to time; we will be
able to solve them by means of their second derivatives
which are vector components. Considering
homogeneity, the right hand terms will have constant coefficients, and
considering isotropy they will be the summations of vector components that were
obtained by differentiation of x, h, z with respect
to x, y, z. But Burkhardt1(Oeuvres
353) has determined all of these vectors. When we admit only the first and
second derivatives, only three exist which are linearly independent, namely
and we will have, a, b, c being constants,
In changing the signs of x, y, z, and
keeping in mind the complete isotropy, we find a = 0. Therefore
only system (15) remains, Q.E.D.
Condition(1) is satisfied by all
mechanical phenomena (and we have seen that in reality it shouldn’t be satisfied
by electromagnetic equations which correspond to irreversible phenomena); (2)
and (3) are satisfied by the phenomena of diffusion, heat propagation, and
others which certainly don’t present any natural connection between them. We
will conclude that the analogy between (14) and (15) is a lot less
characteristic than we would be inclined to believe at first. We won’t
conclude that there’s any real physical connection between the two orders of
phenomena unless the analogy carries over even beyond this general analytical
form. But this is precisely not the case.
Indeed, the hypothesis of the nil speed of propagation for longitudinal waves
(A + B = 0) is not, as Green and Cauchy have previously said, admissible for a
finite elastic body; such a body would not offer any resistance to compression,
its equilibrium would be unstable. It is only recently that Lord Kelvin’s
gyrostatic ether has permitted us to devise such systems. On the other hand,
the hypothesis of incompressibility (Oeuvres 354) calls for the introduction to
the equations one of Lagrange’s factors, performing the role of pressure. The
identification is no longer possible in the cases where the pressure is
constant. Finally, the limiting conditions demanded by Optics are not the same
as in the theory of elasticity.
I don’t believe therefore that we
should consider a priori these analogies as indicative of profound
physical agreement between the two domains. If we adopt this conclusion,
we won’t be overly amazed at the difficulties and strangeness that accompany
all attempts made to extend these analogies of pure ether (where Maxwell’s
equations express only the fact of uniform propagation) to reciprocal actions
of electric charges and the ether, expressed in the general equations (I) thru
(VI). For this part of the question, I can do no more that return to the
chapter which Poincaré has dedicated in his Leçons2 to the
most remarkable, it seems, of these attempts: that of Larmor.
1 BURKHARDT, Math. Annalen, vol.
43, 1893, p. 197; Enzyclop. D. math. Wiss., Bd. 4, Art. 14, 1901, p. 20.
2Électricité et Optique, 2nd
ed., p. 577 and following. – LARMOR, Aether and Matter, Cambridge, 1900.