In these investigations it is necessary to bear in mind that the whole
resisting power of the ether, in disturbing the planetary movements, is
but small, in comparison with gravitation. We will, however, show that,
in the case of the planets, there is a compensation continually made by
this resistance, which leaves but a very small outstanding balance as a
disturbing power. If we suppose all the planets to move in the central
plane of the vortex in circular orbits, and the force of the radial
stream, (or that portion which is not in accordance with the law of
gravitation,) to be inversely as the square roots of the distances from
the sun, it is evident, from what has been advanced, that an equilibrium
could still obtain, by variations in the densities, distances and
diameter of the planets. Supposing, again, that the planets still move
in the same plane, but in elliptical orbits, and that they are in
equilibrium at their mean distances, under the influence or action of
the tangential current, the radial stream, and the density of the ether;
we see that the force of the radial stream is too great at the
perihelion, and too small at the aphelion. At the perihelion the planet
is urged from the sun and at the aphelion towards the sun. The density
and consequent momentum is also relatively too great at the perihelion,
which also urges the planet from the sun, and at the aphelion,
relatively too small, which urges the planet towards sun; and the law is
the same in both cases, being null at the mean distance of the planet,
at a maximum at the apsides; it is, consequently, as the cosine of the
planet's eccentric anomaly at other distances, and is positive or
negative, according as the planet's distance is above or below the mean.
At the planet's mean distance, the circular velocity of the vortex is
equal to the circular velocity of the planet, and, at different
distances, is inversely in the sub-duplicate ratio of those distances.
But the circular velocity of a planet in the same orbit, is in the
simple ratio of the distances inversely. At the perihelion, the planet
therefore moves faster than the ether of the vortex, and at the
aphelion, slower; and the difference is as the square roots of the
distances; but the force of resistance is as the square of the velocity,
and is therefore in the simple ratio of the distances, as we have
already found for the effect of the radial stream, and centrifugal
momentum of the internal ether. At the perihelion this excess of
tangential velocity creates a resistance, which urges the planet towards
the sun, and at the aphelion, the deficiency of tangential velocity
urges the planet from the sun,--the maximum effect being at the apsides
of the orbit, and null at the mean distances. In other positions it is,
therefore, as the cosines of the eccentric anomaly, as in the former
case; but in this last case it is an addititious force at the
perihelion, and an ablatitious force at the aphelion, whereas the first
disturbing force was an ablatitious force at the perihelion, and an
addititious force at the aphelion; therefore, as we must suppose the
planet to be in equilibrium at its mean distance, it is in equilibrium
at all distances. Hence, a planet moving in the central plane of the
vortex, experiences no disturbance from the resistance of the ether.
As the eccentricities of the planetary orbits are continually changing
under the influence of the law of gravitation, we must inquire whether,
under these circumstances, such a change would not produce a permanent
derangement by a change in the mean force of the radial stream, so as to
increase or diminish the mean distance of the planet from the sun. The
law of force deduced from the theory for the radial stream is as the 2.5
power of the distances inversely. But, by dividing this ratio, we may
make the investigation easier; for it is equivalent to two forces, one
being as the squares of the distances, and another as the square roots
of the distances. For the former force, we find that in orbits having
the same major axis the mean effect will be as the minor axis of the
ellipse _inversely_, so that two planets moving in different orbits, but
at the same mean distance, experience a less or greater amount of
centripulsive force from this radial stream, according as their orbits
are of less or greater eccentricity, and this in the ratio of the minor
axis. On the other hand, under the influence of a force acting
centripulsively in the inverse ratio of the square roots of the
distances, we find the mean effect to be as the minor axis of the
ellipse _directly_, so that two planets in orbits of different
eccentricity, but having the same major axis, experience a different
amount from the action of this radial stream, the least eccentric orbit
being that which receives the greatest mean effect. By combining these
two results, we get a ratio of equality; and, consequently, the action
of the radial stream will be the same for the same orbit, whatever
change may take place in the eccentricity, and the mean distance of the
planet will be unchanged. A little consideration will also show that the
effect of the centrifugal momentum due to the density of ether will also
be the same by change of eccentricity; for the positive will always
balance the negative effect at the greatest and least distances of the
planet. The same remark applies to the effect of the tangential current,
so that no change can be produced in the major axes of the planetary
orbits by change of eccentricity, as an effect of the resistance of the
ether.
We will now suppose a planet's orbit to be inclined to the central plane
of the vortex, and in this case, also, we find, that the action of the
radial stream tends to increase the inclination in one quadrant as much
as it diminishes it in the next quadrant, so that no change of
inclination will result. But, if the inclination of the orbit be changed
by planetary perturbations, the mean effect of the radial stream will
also be changed, and this will tell on the major axis of the orbit,
enlarging the orbit when the inclination diminishes, and contracting it
when it increases. The change of inclination, however, must be referred
to the central plane of the vortex. Notwithstanding the perfection of
modern analysis, it is confessed that the recession of the moon's nodes
does yet differ from the theory by its 350th part, and a similar
discrepancy is found for the advance of the perigee.[40] This theory is
yet far too imperfect to say that the action of the ethereal medium will
account for these discrepancies; but it certainly wears a promising
aspect, worthy the notice of astronomers. There are other minute
discordancies between theory and observation in many astronomical
phenomena, which theory _is_ competent to remove. Some of these we shall
notice presently; and, it may be remarked, that it is in those minute
quantities which, in astronomy, are usually attributed to errors of
observation, that this theory will eventually find the surest evidence
of its truth.