will be the very small arc De that the satellite describes in a second. If the attractive force of the planet were to cease for an instant, the satellite would fly off in the tangent De, and would be farther from the centre of the planet by a quantity equal to aD, the versed sine of the arc Dc. But the value of the versed sine is

which is the distance that the attractive force of the planet causes the satellite to fall through in a second.

Now, if another satellite be considered, whose mean distance is Pd a', and T, the duration of its sidereal revolution, its deflection

=

will be a'

2T2

in a second; but if F and F be the attractive forces T

of the planet at the distances PD and Pd, they will evidently be proportional to the quantities they make the two satellites fall through in a second;

but the squares of the periodic times are as the cubes of the mean distances; hence

Thus the satellites gravitate to their primaries inversely as the square of the distance.

325. As the earth has but one satellite, this comparison cannot be made, and therefore the ellipticity of the lunar orbit is the only celestial phenomenon by which we can know the law of the moon's attractive force. If the earth and the moon were the only bodies in the system, the moon would describe a perfect ellipse about the earth; but, in consequence of the action of the sun, the path of the moon is sensibly disturbed, and therefore is not a perfect ellipse; on this account some doubts may arise as to the diminution of the attractive force of the earth as the inverse square of the distance.

The analogy, indeed, which exists between this force and the attractive force of the sun, Jupiter, and Saturn, would lead to the belief that it follows the same law, because the solar attraction acts equally on all bodies placed at the same distance from the sun, in the same manner that terrestrial gravitation causes all bodies in vacuo to fall from equal heights in equal times. A projectile thrown horizontally from a height, falls to the earth after having described a parabola. If the force of projection were greater, it would fall at a greater distance; and if it amounted to 30772.4 feet in a second, and were not resisted by the air, it would revolve like a satellite about the earth, because its centrifugal force would then be equal to its gravitation. This body would move in all respects like the moon, if it were projected with the same force, at the same height.

It may be proved, that the force which causes the descent of heavy bodies at the surface of the earth, diminished in the inverse ratio of the square of the distance, is sufficient to retain the moon in her orbit, but this requires a knowledge of the lunar parallax.

On Parallax.

326. Let m, fig. 65, be a body in its orbit, and C the centre

an observer at E, the angle zEm, called the zenith distance of the body, may be measured; hence mEC is known, and the difference between zEm and zCm is equal to CmE, the parallax, then if CER, Cmr, and zEm = 2,

hence, if CE and Cm remain the same, the sine of the parallax, CmE, will vary as the sine of the zenith distance zEm; and when zEm 90°, as in fig. 67,

P being the value of the angle CmE in this case; then the parallax is a maximum, for Em is tangent to the earth, and, as the body m is seen in the horizon, it is called the horizontal parallax; hence the sine of the horizontal parallax is equal to the terrestrial radius divided by the distance of the body from the centre of the earth.

327. The length of the mean terrestrial radius is known, the horizontal parallax may be determined by observation, therefore the distance of m from the centre of the earth is known. By this method the dimensions of the solar system have been ascertained with great accuracy. If the distance be very great compared with the diameter of the earth, the parallax will be insensible. If CmE were an angle of the fourth of a second, it would be inappreciable; an arc of 1" = 0.000004848 of the radius, the fourth of a second is there1 fore 0.000001212 = ; and thus, if a body be distant from 825082

the earth by 825082 of its semidiameters, or 3265660000 miles, it will be seen in the same position from every point of the earth's surface. The parallax of all the celestial bodies is very small: even that of the moon at its maximum does not much exceed 1o.

328. P being the horizontal parallax, let p be the parallax EmC, fig. 66, at any height. When P is known, p may be found, and the contrary, for if

R

r

be eliminated, then sin p sin P sin z, and

when P is constant, sin p varies as sin z.

329. The horizontal parallax is determined as follows: let E and E', fig. 66, be two places on the

same meridian of the earth's surface; fig.66.
that is, which contemporaneously have
the same noon. Suppose the latitudes
of these two places to be perfectly
known; when a body m is on the me-
ridian, let its zenith distances

zEm2, z'E'm = z',

be measured by two observers in E and E'. Then ECE', the sum of the lati

m

E

E

tudes, is known, and also the angles CEm, CE'm; hence EmE', EmC, and E'mC may be determined; for P is so small, that it may be put for its sine; therefore

sin p = P sin z, sin p' = P sin z' ;

and as p and p' are also very small,

p+ p' = P {sin z + sin z'.}

Now, p + p' is equal to the angle EmE', under which the chord of the terrestrial arc EE', which joins the two observers, would be seen from the centre of m, and it is the fourth angle of the quadrilateral CEmE'.

which is the horizontal parallax of the body, when the observers are on different sides of Cm; but when they are on the same side,

It requires a small correction, since the earth, being a spheroid, the lines ZE, Z'E' do not pass through C, the centre of the earth.

The parallax of the moon and of Mars were determined in this manner, from observations made by La Caille at the Cape of Good Hope, in the southern hemisphere; and by Wargesten at Stock

holm, which is nearly on the same meridian in the northern hemisphere.

330. The horizontal parallax varies with the distance of the body from the earth; for it is evident that the greater the distance, the less the parallax. It varies also with the parallels of terrestrial latitude, the earth, being a spheroid, the length of the radius decreases from the equator to the poles. It is on this account that, at the mean distance of the moon, the horizontal parallax observed in different latitudes varies; proving the elliptical figure of the earth. The difference between the mean horizontal parallax at the equator and at the poles, from this cause, is 10".3.

331. In order to obtain a value of the moon's horizontal parallax, independent of these inequalities, the horizontal parallax is chosen at the mean distance of the moon from the earth, and on that parallel of terrestrial latitude, the square of whose sine is, because the attraction of the earth upon the corresponding points of its surface is nearly equal to the mass of the earth, divided by the square of the mean distance of the moon from the earth. This is called the constant part of the horizontal parallax. The force which retains the moon in her orbit may now be determined.

Force of Gravitation at the Moon.

332. If the force of gravity be assumed to decrease as the inverse

m

fig. 67.

square of the distance, it is clear that the force of gravity at E, fig. 67, would be, to the same force at m, the distance of the moon, as the square of C'm to the square of CE; but CE divided by Cm is the sine of the horizontal parallax of the moon, the constant part of which is found by observation to be 57' 4". 17 in the latitude in question; hence the force of gravity, reduced to the distance of the moon, is equal to the force of gravity at E on the earth's surface, multiplied by sin° 57′ 4′′. 17, the square of the sine of the constant part of the horizontal parallax.

Since the earth is a spheroid, whose equatorial diameter is greater than its polar diameter, the force of gravity increases from the equa