distance. Then, since the whole force of gravity at a given distance is proportional to the mass, we have mf force of gravity of the mass m at a unit of distance. Hence, g being taken for its force at the distance d, we have,

In like manner for the sun and planet, the mass of the sun

In this investigation, the attraction of the planet on the sun, and that of the satellite on the planet, have both been omitted. But as the mass of the planet is very small in comparison with that of the sun, and the mass of the satellite in comparison with that of the planet, the result is but little affected by the omission. We have, thus, a very simple formula for computing, with considerable accuracy, the mass of a planet that is attended by a satellite.

Applying this formula to the planet Neptune, we have (335 and 336), P = 1641 years, p= 5 days, 21 hours, D= 2,850,000,000 miles, and d = 230,000 miles, which give, for the mass of Neptune, m = 18000 nearly. This result accords very well with the value given in Article 336.

The masses of the planets which have no satellite, and also that of the moon, are deduced, by more difficult investigations, from. the ascertained effects of their actions on other bodies.

401. The Densities of the Sun and Planets. The densities of bodies are proportional to their masses, divided by their volumes.

Hence, from the known masses and volumes of the sun and planets, their densities are easily obtained.*

402. The density of the earth increases towards the centre.

Supposing the earth to have been once in a fluid state and homogeneous throughout, it is ascertained by investigation that, in consequence of its revolution on its axis, it would have taken the form of an oblate spheroid, having the polar radius to the equatorial in the ratio of 229 to 230, and, consequently, have an ellipticity of 20. If, instead of being homogeneous, it is composed of strata increasing in density towards the centre, the form would still be that of an oblate spheroid, but of less ellipticity. Hence, as the actual ellipticity of the earth, which is only do, is considerably less than 30, and as it is probable the earth was once in the state supposed, it is inferred that the density increases towards the

centre.

1

1 300

This inference is confirmed by very accurate observations made at the sides of the mountain Schehallion in Scotland, by Dr. Maskelyne. From the effect of the mountain in changing the direction of the plumb line of a plummet suspended near it, and from the known figure and volume of the mountain determined by a survey, it was found that the mean density of the mountain was to that of the whole earth, nearly as 5 to 9.

403. Kepler's Laws, though very nearly, are not rigorously true. The deviation from entire accuracy is caused by the attractions of the planets on the sun and on one another, and also by the attractions of the satellites on their primaries. But, as the masses of all the planets taken together are very small in comparison with that of the sun, and those of the satellites in comparison with those of their primaries, the deviation with regard to either of the laws is also small.

404. The sun's action increases the gravity of the moon to the earth at the quadratures, and diminishes it twice as much at the syzygies; the effect, on the whole, being a diminution of her gravity to the earth by about the 358th part.

* The masses and densities of the sun, planets and moon, as deduced from the most accurate investigations, are given in the tables at the end of this part of the

work.

Let ACBO, Fig. 57, represent the orbit of the moon, which may in this investigation be considered as coinciding with the plane of the ecliptic. Also let S be the sun, E the earth, M the place of the moon in her orbit, and AB perpendicular to SE, the line of the quadratures. Let the line SE represent the force which the sun exerts on the earth at E, or on the moon, when in quadratures,

at A and B.* Then, SM2: SE2 :

SE:

SE3
SM2

the force with

which the sun acts on the moon at M. In the line MS, produced

SE3

if necessary, take MD SM2; then MD represents the force

which the sun exerts on the moon at M. Let the force MD be resolved into the two, MH and MG, one of which, MH, is equal and parallel to ES. Then since the force MH is equal and parallel to ES, it will have no tendency to change the relative motions or positions of the earth and moon. The other force MG, will therefore represent, in quantity and direction, the whole effect of the sun's action in disturbing the moon's motion in her orbit. Let SM be produced to meet the diameter AB in N. Then, because the angle ESN is always very small, being when greatest only about 7', the line SN may be considered equal to SE. Hence,

SM3 + 3SM2 × MN + 3SM × MN2 + MN3

But, as MN is very small in comparison with SM, the last two terms may be omitted without material error.

Therefore, MD = SM + 3MN; or, SD = 3MN.

As the angle ESM is very small, and SD is also small, the line DG must very nearly coincide with SE, and, consequently, the

Strictly speaking, as the quantity of matter in the earth is greater than that in the moon, the forces which the sun exerts on the earth and moon, when at equal distances, are not equal. But the effects of those forces, in moving the bodies, are equal, and it is these effects which is the subject under consideration.

point G with the point L. We may therefore consider ML as the force by which the sun disturbs the motion of the moon. Now, EL + LS=ES=HM-DG-SD + LS, very nearly, or, EL-SD, very nearly.

Hence, if MK be perpendicular to SE, we have,

EL 3MN 3EK.

Let the force ML be resolved into two others, one MQ in the direction of the radius vector, and the other MP in the direction of a tangent to the orbit at M. Then the force MQ increases or diminishes the gravity of the moon to the earth, according as the point Q falls between E and M, or in EM produced. The other force MP increases or diminishes the moon's angular motion about the earth. Since the moon's orbit does not differ much from a circle, the angle QMP may be considered as a right angle. Put α SE, r EM, x = the angle AEM, m = mass of the earth, that of the sun being 1, and ƒ force of gravity of the sun, or f of a unit of mass, at a unit of distance. Then,

Or, using the affirmative sign to denote an increase in the moon's gravity to the earth.

MQ = (1-3 sin x).
+ r

Now the analytical expression of the force represented by ES,

f

is evidently Hence, since ES: MQ :: the force ES: the

a2*

When the moon is in quadratures, x = 0, or 180°. Consequently, then,

the force MQ +

fr

a3

.(K)

But when the moon is in syzygies, x = 90° or 270°. Hence, then,

The first part of the proposition is, therefore, proved.

Now, it is evident, from (H), that the force MQ = 0, when 3 sin2x 1, or sin x = ✓ }; that is, when x 35° 15′52′′. The moon's gravity to the earth is, therefore, increased while she is within about 35° of her quadratures, on either side, and is diminished in all the remaining part of the orbit; and the greatest diminution is double the greatest increase. It follows, therefore, that in the whole the moon's gravity to the earth is diminished by the action of the sun. An easy investigation, with the aid of differential calculus, proves that the mean or average diminution is r representing in this case the mean distance of the moon from the earth.

fr

2a3,

fr 2a39

Dividing the mean diminution of the moon's gravity to the

earth by mf, which expresses the whole gravity, the quotient

203

2ma3,

is the mean diminution of the moon's gravity expressed as a fraction of the whole. Substituting, in this, the value of m (391 G), and observing that a and r are used here in the place of D and d, we have,

405. The inequality in the moon's motion, called the Annual Equation (204), proceeds from an inequality in the sun's disturbing force, depending on the variation in the earth's distance from the sun.

The expression

fr

2a3

designates the mean diminution of the moon's