From this expression the masses of such planets as have satellites may be obtained.

Suppose this equation relative to the earth, and that the mass of the earth is omitted when compared with that of the sun, it then becomes

Again, let μm + m' the sum of the masses of a planet and of its satellite m', T' being the time of a sidereal revolution of the planet at the mean distance a' from the sun, then

If the values of T, T', a and a', determined from observation, be substituted in this expression, the ratio of the sum of the masses of the planet and of its satellite to the mass of the sun will be obtained; and if the mass of the satellite be neglected when compared with that of its primary, or if the ratio of these masses be known, the preceding equation will give the ratio of the mass of the planet to that of the sun. For example,

599. Let m be the mass of Jupiter, that of his satellite being omitted, and let the mass of the sun be taken as

fig. 91.

Jm the mean radius of the orbit of the fourth satellite at the mean distance of the earth from the sun taken as the unit, is seen under the angle JEm 2580".579

The radius of the circle reduced to seconds is 206264".8; hence the mean radii of the orbit of the fourth satellite and of the terrestrial orbit are in the ratio of these two numbers. The time of a sidereal revolu

tion of the fourth satellite is 16.6890 days, and the sidereal year is 365.2564 days, hence

With these data it is easy to find that the mass of Jupiter is

m=

1

1066.09

The sixth satellite of Saturn accomplishes a sidereal revolution in 15.9453 days; the mean radius of its orbit, at the mean distance of the planet, is seen from the sun under an angle of 179"; whence the 1 mass of Saturn is 3359.40

By the observations of Sir William Herschel the sidereal revolutions of the fourth satellite of Uranus are performed in 13.4559 days, and the mean radius of its orbit seen from the sun at the mean distance of the planet is 44".23. With these data the mass of Uranus

600. This method is not sufficiently accurate for finding the mass of the Earth, on account of the numerous inequalities of the Moon. It has already been observed, that the attraction of the Earth on bodies at its surface in the parallel where the square of the sine of the latitude is, is nearly the same as if its mass were united at its centre of gravity. If R be the radius of the terrestrial spheroid drawn to that parallel, and m its mass, this attraction will be

Then, if a be the mean distance of the Sun from the Earth, T the duration of the sidereal year,

R, g, T, and a, are known by observation, therefore the ratio of the

mass of the Earth to that of the Sun may be found from this expression.

The sine of the solar parallax at the mean distance of the sun from the earth, and in the latitude in question, is

the attraction of the Earth, and the terrestrial radius in the same

with these data the mass of the earth is computed to be

1 337103

the mass of the sun being unity. This value varies as the cube of the solar parallax compared with that adopted.

601. The compression of the three larger planets, and the ring of Saturn, probably affect the values of the masses computed from the elongations of their satellites; but the comparison of numerous well chosen observations, with the disturbances determined from theory, will ultimately give the masses of all the planets with great accuracy.

The action of each disturbing body adds a term of the form m'v' to the longitude, so that the longitude of m at any given instant in its troubled orbit, is

v + m'dv' + m''Sv" + &c. v1, dv', dv", &c.

are susceptible of computation from theory; and as they are given by the Tables of the Motions of the Planets, the true longitude of m is v + m'dv' + m"Sv" + &c. = L.

When this formula is composed with a great number of observations, a series of equations,

are obtained, where m', m", &c., are unknown quantities, and by the resolution of these the masses of the planets may be estimated by the perturbations they produce.

602. As there are ten planets, ten equations would be sufficient to

give their masses, were the observed longitudes and the computed quantities v, dv', dv", &c., mathematically exact; but as that is far from being the case, many hundreds of observations made on all the planets must be employed to compensate the errors. The method of combining a series of equations more numerous than the unknown quantities they contain, so as to determine these quantities with all possible accuracy, depends on the theory of probabilities, which will be explained afterwards. The powerful energy exercised by Jupiter on the four new planets in his immediate vicinity occasions very great inequalities in the motions of these small bodies, whence that highly distinguished mathematician, M. Gauss, has obtained a value for the mass of Jupiter, differing considerably from that deduced from the elongation of his satellites, it cannot however be regarded as conclusive till the perturbations of these small planets are perfectly known.

603. The mass of Venus is obtained from the secular diminution in the obliquity of the Ecliptic. The plane of the terrestrial equator is inclined to the plane of the ecliptic at an angle of 23° 28′ 47′′ nearly, but this angle varies in consequence of the action of the planets. A series of tolerably correct observations of the Sun's altitude at the solstices chiefly by the Chinese and Arabs, have been handed down to us from the year 1100 before Christ, to the year 1473 of the Christian era; by a comparison of these, it appears that the obliquity was then diminishing, and it is still decreasing at the rate of 50.2 in a century. From numerous observations on the obliquity of the ecliptic made by Bradley about a hundred years ago, and from later observations by Dr. Maskelyne, Delambre determined the maximum of the inequalities produced by the action of Venus, Mars, and the Moon, on the Earth, and by comparing these observations with the analytical formulæ, he obtained nearly the same value of the mass of Venus, whether he deduced it from the joint observations of Bradley and Maskelyne, or from the observations of each separately. From this correspondence in the values of the mass of Venus, obtained from these different sets of observations, there can be little doubt that the secular diminution in the obliquity of the ecliptic is very nearly 50".2, and the probability of accuracy is greater as it agrees with the observations made by the Chinese and Arabs so

many centuries ago. Notwithstanding doubts still exist as to the mass of Venus.

604. The mass of Mars has been determined by the same method, though with less precision than that of Venus, because its action occasions less disturbance in the Earth's motions, for it is evident that the masses of those bodies that cause the greatest disturbance will be best known. The action of the new planets is insensible, and that of Mercury has a very small influence on the motions of the rest. An ingenious method of finding the mass of that planet has been adopted by La Place, although liable to error.

605. Because mass is proportional to the product of the density and the volume, if m, m', be the masses of any two planets of which P, p', are the densities, and V, V, the volumes, then

But as the planets differ very little from spheres, their volumes may be assumed proportional to the cubes of their diameters; hence if D, D', be the diameters of m, and m',

The apparent diameters of the planets have been measured so that D and D' are known; this equation will therefore give the densities if the masses be known, and vice versâ.

By comparing the masses of the Earth, Jupiter, and Saturn, with their volumes, La Place found that the densities of these three planets are nearly in the inverse ratio of their mean distances from the sun, and adopting the same hypothesis with regard to Mercury, Mars, and Jupiter, he obtained the preceding values of the masses of Mars and Mercury, which are found nearly to agree with those determined from other data. Irradiation, or the spreading of the light round the disc of a planet, and other difficulties in measuring the apparent diameters, together with the uncertainty of the hypothesis of the law of the densities, makes the values of the masses obtained in this way the more uncertain, as the hypothesis does not give a true result for the masses of Venus and Saturn. Fortunately the influence of Mercury on the solar system is very small.