(u,v,w) at the point (x,y,z), is simply superposed on the
velocity V of the optical undulations through that medium,
the latter not being intrinsically altered. Now the direction
and phase of the light are those of the ray which reaches the
eye; and by Fermat's principle, established by Huygens for
undulatory motion, the path of a ray is that track along which
the disturbance travels in least time, in the restricted sense
that any alteration of any short reach of the path will
increase the time. Thus the path of the ray when the aether is
at rest is the curve which makes Integralds/V least; but when it is
in motion it is the curve which makes Integralds/(V+lu+my+nw) least,
where (l,m,n) is the direction vector of ds.
The latter integral becomes, on expanding in a series,
Integralds/V - Integral(udx + vdy + wdz)/V2 + Integral(udx + vdy + wdz)2/V3 + ...,
since lds=dx. If the path is to be unaltered by the motion
of the aether, as the law of astronomical aberration suggests,
this must differ from Integralds/V by terms not depending on the
path--that is, by terms involving only the beginning and end of
it. In the case of the free aether V is constant; thus, if
we neglect squares like (u/V)2, the condition is that udx
+ vdy + wdz be the exact differential of some function f.
If this relation is true along all paths, the velocity of the
aether must be of irrotational type, like that of frictionless
fluid. Moreover, this is precisely the condition for the
absence of interference between the component of a split
beam; because, the time of passage being to the first order
Integralds/V - Integral(udx + vdy + wdz)/V2
the second term will then be independent of the path (f
being a single valued function) and therefore the same for the
paths of both the interfering beams. If therefore the aether
can be pnt into motion, we conclude (with Stokes) that such
motion, in free space, must be of strictly irrotational type.
But our experimental data are not confined to free space. if
c is the velocity of radiation in free space and m the
refractis'e index of a transparent body, V=C/m; thus it
is the expression c-2Integralm2(u'dx + v'dy + w'dz) that
is to be integrable explicitly, where now (u',v',w') is
what is added to V owing to the velocity (u,v,w) of the
medium. As, however, our terrestrial optical apparatus is
now all in motion along with the matter, we must deal with
the rays relative to the moving system, and to these also
Fermat's principle clearly applies; thus V + (lu' + mv' +
nw') is here the velocity of radiation in the direction of
the ray, but relative to the moving material system. Now the
expression above given cannot be integrable exactly, under
all circumstances and whatever be the axes of co-ordinates,
unless (m2u',m2v',m2w') is the gradient of a continuous
function. In the simplest case, that of uniform translation,
these components of the gradient will each be constant
throughout the region; at a distant place in free aether where
there is no motion, they must thus be equal to -u,-v,
-w, as they refer to axes moving with the matter. Hence
the paths and times of passage of all rays relative to the
material system will not be altered by a uniform motion of the
system, provided the velocity of radiation relative to the
system, in material of index m, is diminished by m-2
times the velocity of the system in the direction of the
radiation, that is, provided the absolute velocity of radiation
is increased by 1 - m-2 times the velocity of the material
system; this involves that the free aether for which m is
unity shall remain at rest. This statement constitutes the
famous hypothesis of Fresnel, which thus ensures that all
phenomena of ray-path and refraction, and all those depending
on phase, shall be unaffected by uniform convection of the
material medium, in accordance with the results of experiment.
Is the Aether Stationary or mobile?---This theory secures
that the times of passage of the rays shall be independent of
the motion of the system, only up to the first order of the
ratio of its velocity to that of radiation. But a classical
experiment of A. A. Michelson, in which the ray-path was
wholly in air, showed that the independence extends to higher
orders. This result is inconsistent with the aether remaining at
rest, unless we assume that the dimensions of the moving system
depend, though to an extent so small as to be not otherwise
detectable, on its orientation with regard to the aether that
is streaming through it. It is, however, in complete accordance
with a view that would make the aether near the earth fully
partake in its orbital motion---a view which the null effect of
convection on all terrestrial optical and electrical phenomena
also strongly suggests. But the aether at a great distance
must in any case be at rest; while the facts of astronomical
aberration require that the motion of that medium must be
irrotational. These conditions cannot be consistent with
sensible convection of the aether near the earth without
involving discontinuity in its motion at some intermediate
distance, so that we are thrown back on the previous theory.
Another powerful reason for taking the aether to be
stationary is afforded by the character of the equations of
electrodynamics; they are all of linear type, and superposition
of effects is possible. Now the kinetics of a medium in
which the parts can have finite relative motions will lead
to equations which are not linear---as, for example, those of
hydrodynamics---and the phenomena will be far more complexly
involved. It is true that the theory of vortex rings in
hydrodynamics is of a simpler type; but electric currents
cannot be likened to permanent vortex rings, because their
circuits can be broken and the element of cyclic steadiness
on which the simplicity depends is thereby destroyed.
Dynamical Theories of the Aether.---The analytical equations
which represent the propagation of light in free aether,
and also in aether modified by the presence of matter, were
originally developed on the analogy of the equations of
propagation of elastic effects in solid media. Various types
of elastic solid medium have thus been invented to represent
the aether, without complete success in any case. In T.
Maccullagh's hands the correct equations were derived from a
single energy formula by the principle of least action; and
while the validity of this dynamical method was maintained,
it was frankly admitted that no mechanical analogy was
forthcoming. When Clerk Maxwell pointed out the way to the
common origin of optical and electrical phenomena, these
equations naturally came to repose on an electric basis, the
connexion having been first definitely exhibited by Fitzgerald
in 1878; and according as the independent variable was one or
other of the vectors which represent electric force, magnetic
force or electric polarity, they took the form appropriate
to one or other of the elastic theories above mentioned.
In this place it must suffice to indicate the gist of the
more recent developments of the electro-optical theory, which
involve the dynamical verification of Fresnel's hypothesis
regarding optical convection and the other relations above
described. The aether is taken to be at rest; and the
strain-forms belonging to the atoms are the electric fields
of the intrinsic charges, or electrones, involved in their
constitution. When the atoms are in motion these strain-forms
produce straining and unstraining in the aether as they pass
across it, which in its motional or kinetic aspect constitutes
the resulting magnetic field; as the strains are slight
the coefficient of ultimate inertia here involved must be
great. True electric current arises solely from convection
of the atomic charges or electrons; this current is therefore
not restricted as to form in any way. But when the rate of
change of aethereal strain----that is, of (f,g,h) specified
as Maxwell's electric displacement in free aether---is
added to it, an analytically convenient vector (u,v,w)
is obtained which possesses the characteristic property of
being circuital like the flow of an incompressible fluid,
and has therefore been made fundamental in the theory
by Maxwell under the name of the total electric current.
As already mentioned, all efforts to assimilate optical
propagation to transmission of waves in an ordinary solid
medium have failed; and though the idea of regions of
intrinsic strain, as for example in unannealed glass,
is familiar in physics, yet on account of the absence of
mobility of the strain no attempt had been made to employ
them to illustrate the electric fields of atomic charges.
The idea of Maccullagh's aether, and its property of purely
rotational elasticity which had been expounded objectively
by W. J. M. Rankine, was therefore much vivified by Lord
Kelvin's specification (Comptes Rendus, 1889) of a material
gyrostatically constituted medium which would possess this
character. More recently a way has been pointed out in which
a mobile permanent field of electric force could exist in such
a medium so as to travel freely in company with its nucleus or
intrinsic charge---the nature of the mobility of the latter,
as well as its intimate constitution, remaining unknown.
A dielectric substance is electrically polarized by a field
of electric force, the atomic poles being made up of the
displaced positive and negative intrinsic charges in the
atom: the polarization per unit volume (f',g',h') may be
defined on the analogy of magnetism, and d/dt(f', g', h')
thus constitutes truo electric current of polarization,
i.e. of electric separation in the molecules, specified
per unit volume. The convection of a medium thus polarized
involves electric disturbance, and therefore must contribute
to the true electric current; the determination of this
constituent of the current is the most delicate point in the
investigation. The usual definition of the component current
in any direction, as the net amount of electrons which crosses,
towards the positive side, an element of surface fixed in
space at right angles to that direction, per unit area per
unit time, here gives no definite result. The establishment
and convection of a single polar atom constitutes in fact a
quasi-magnetization, in addition to the polarization current
as above defined, the negative poles completing the current
circuits of the positive ones. But in the transition from
molecular theory to the electrodynamics of extended media,
all magnetism has to be replaced by a distribution of current;
the latter being now specified by volume as well as by flow
so that (u,v,w) dt is the current in the element of volume
dt. In the present case the total dielectric contribution
to this current works out to be the change per unit time in
the electric separation in the molecules of the element of
volume, as it moves uniformly with the matter, all other
effects being compensated molecularly without affecting the
propagation.1 On subtracting from this total the current of
establishment of polarization d/dt/(f,g',h') as formulated
above, there remains vd/dx(f',g',h') as the current of
convection of polarization when the convection is taken
for simplicity to be in the direction of the axis of x
with velocity v. The polarization itself is determined
from the electric force (P,Q,R) by the usual statical
formula of linear type which becomes tor an isotropic medium
(f',g',h') = ((K-1)/4pc2)(P,Q,R),
because any change of the dielectric constant K arising
