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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
sin P = = sin 8".75;
the attraction of the Earth, and the terrestrial radius in the same
and the sidereal
g= 2.16.1069 = 32.2138
with these data the mass of the earth is computed to be
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"dv" + &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
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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 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
m: m' :: p
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',
m: m' :: p. D3 : p' . D13 ;
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 versa.
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.
606. The mass of the Sun being unity, the masses of the planets are,
Densities of the Planets.
607. The densities of bodies are proportional to the masses divided by the volumes, and when the masses are spherical, their volumes are as the cubes of their radii; as the sun and planets are nearly spherical, their densities are therefore as their masses divided by the cubes of their radii; but the radii must be taken in those parallels of latitude, the squares of whose sines are .
The mean apparent semidiameters of the Sun and Earth at their mean distance are,
The radius of Jupiter's spheroid in the latitude in question, when viewed at the mean distance of the earth from the sun, is 94".344 ; and the corresponding radius of Saturn at his mean distance from the sun is 8".1. Whence the densities are,
Thus the densities decrease with the distance from the sun; however that of Uranus does not follow this law, being greater than that of Saturn, but the uncertainty of the value of its apparent diameter may possibly account for this deviation.
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Intensity of Gravitation at the Surfaces of the Sun and
608. Let g and g' represent the force of gravity at the surfaces of two bodies m and m', whose apparent diameters are D and D'. If the bodies be spherical and without rotation, the force of gravity at their equators will be as their masses divided by the squares of their diameters;
Because the masses, apparent diameters, and the intensity of gravity at the terrestrial equator are known, g, the intensity of the gravitating force at the equator of any other body may be found; and as the rotation of the sun and planets is determined by observation, their centrifugal forces, and consequently the intensity of gravitation at their surfaces may be computed. With the preceding values of the masses and apparent diameters it will be found, that if the weight of a body at the terrestrial equator be the unit, the same body transported to the equator of Jupiter, would weigh 2.716; but this would be diminished by about a ninth, on account of the centrifugal force. The same body would weigh 27.9 at the sun's equator, and a body at the sun's equator would fall through 448.39 feet in the first second of its descent, that would only fall through 16.0436 feet at the earth's equator.
To determine the fall of bodies at the surfaces of the sun and planets was hopeless till Newton's immortal discovery connected us with remote worlds.
609. The mean sidereal motions of the planets in a Julian year of 365.25 days are the second data.
When the sun is in the tropics his declination is a maximum, and equal to the obliquity of the ecliptic; the time at which that happens is found by observing his declination at noon for several days before and after the instant of a solstice, so that an equation can be formed between the time and the declination, which is sufficiently exact for a few days. If the differential of the declination in this equation be made zero, the instant of the solstice and the obliquity of