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and if the ecliptic of 1750 be assumed as the fixed plane = 0: v' is the heliocentric longitude of the earth.
Let √ddy's+dz', the little are described by the earth in the time dt be represented by r'ds'. This arc is to that described by
the moon in her relative motion round the earth as to unity, con
sequently at least thirty times as great. If the eccentricity of the terrestrial orbit be omitted, ds' = mdt. If these quantities be substituted for the co-ordinates
ma'dt- dx. sin v' + dy. cos v';
and if quantities depending on the arc 2v be rejected,
The different quantities contained in this equation must now be determined.
789. The distance of the moon from the earth is Em =
1, that of
but is a very small fraction that may be omitted; consequently,
when the square root is extracted, the distance of the moon from the sun is
If we assume the density of the ether to be proportional to a function of the distance from the sun, and represent that function by (u'), with regard to the moon, it will be
quantity depending on the density of the ether it is variable, hence it
790. By the substitution of these quantities in equation (241) it will be found, after rejecting periodic quantities, and integrating, that
which is the secular variation in the mean parallax of the moon in consequence of the resistance of the ether.
In order to abridge, let
α = Hma
The value of
= − av + 6. e sin (cv).
in equation (225) will be augmented by av, there
Consequently, when periodic quantities are omitted, =-3fadv.dR
Thus the mean motion is affected by a secular variation from the resistance of the ethereal medium; but it may easily be shown, from the value of R in article 788, that this medium has no effect whatever on the motion of the lunar nodes or perigee. However, in consequence of that action the second of equations (224), which is the coefficient of sin (cv), ought to be augmented by 6.e; hence, rejecting c2, do, and making c = 1 it gives
Thus the eccentricity of the lunar orbit is affected by a secular inequality from the resistance of ether, but it is insensible when compared with the corresponding inequality in the mean motion.
It appears then that the mean motion of the moon is subject to a secular variation in consequence of the resistance of ether, which neither affects the motion of the perigee nor the position of the orbit; and, as the secular inequalities of the moon deduced theoretically
from the variation of the eccentricity of the earth's orbit are perfectly confirmed by the concurrence of ancient and modern observations, they cannot be ascribed to the resistance of an ethereal medium. 791. The action of the ether on the motions of the earth may be found by the preceding formulæ to be
when the eccentricity of the earth's orbit is omitted, so that
If (u') be a function of the distance of the earth from the moon, then must K' H'. p (u'), H' being a constant quantity depending on the mass and surface of the earth. Whence it may be found by the same method with that employed, that the resistance of ether in the mean motion of the earth would be
H'a'm1t2 .$ (u').
Whence it appears that the acceleration in the mean motion of the moon is to that in the mean motion of the earth as unity to
Now H' and H depend on the masses and surfaces of the earth and moon; and as the resistance is directly as the surface, and inversely as the mass, therefore
But by article 652, if the radius of the earth be unity, the moon's true diameter =
But as the terrestrial radius is assumed = 1, the earth's surface is
mass of moon
H mass of earth
square horizontal parallax of moon square of moon's apparent diameter
From observation half the moon's apparent diameter is 943′′.164, her horizontal parallax is 3454.16, and her mass is of that of the
earth, so = 0.17883; and as m =
H acceleration in the mean motion of the earth from the resistance of ether is equal to the corresponding acceleration in the mean motion of the moon multiplied by 0.008942, or about a hundred times less than the acceleration of the moon from the resistance of ether. No such acceleration has been detected in the earth's motion, nor could it be expected, since it is insensible with regard to the moon.
In the preceding investigation, the resistance was assumed to be as the square of the velocity, but Mr. Lubbock has obtained general formula, which will give the variations in the elements, whatever the law of this resistance may be.
792. Although we have no reason to conclude that the sun is surrounded by ether, from any effects that can be ascribed to it in the motions of the moon and planets, the question of the existence of such a fluid has lately derived additional interest from the retardation that has been observed in the returns of Enke's comet at each revolution, which it is difficult to account for by any other supposition than this existence of such a medium.
Mr. Enke has proved that this retardation does not arise from the disturbing action of the planets. But on computing