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Planets

earth, sun, mean, distance, masses, jupiter and divided

PLANETS, masses of It would appear, at first view, impossible to ascertain the respective masses of the Sun and planets, and to calculate the velocity with which heavy bodies fall towards each when at a given distance from their centres ; yet these points may be determined from the theory of gravitation without much diffi culty. It follows, however, from certain theorems relative to centrifugal forces, that the gravitation of a satellite to wards its planet is to the gravitation of lie Earth towards the Sun, as the mean distance of the satellite from its prima. ry, divided by the square of the time of its sidereal revolution, or the mean distance of the Earth from the Sun di vided by the square of a sidereal year. To bring these gravitations to the same distance from the bodies which produce them, we Must multiply them respective ly by the squares of the radii of the orbits which are described : and, as at equal distances the masses are proportional to the attractions, the mass of the Earth is to that of the Sun as the cube of the mean radius of the orbit of the satel lite, divided by the square of the time of its sidereal motion, is to the cube of the mean distance of the Earth from the Sun, divided by the square of the side real year. Let us apply this result to Jupiter. The mean distance of his fourth satellite subtends an angle of 1530".86 decimal seconds. Seen at the mean dis tance of the Earth from the Sun, it would appear under an angle of 7964" .75 deci mal seconds. The radius of the circle contains 636,619" .8 decimal seconds. Therefore the mean radii of the orbit of Jupiter's fourth satellite, and of the Earth's orbit, are to each other as these two numbers. The time of the sidereal revolution of the fourth satellite is 16.6890 days ; the sidereal year is 365.2564 days.

These data give us1066.08for the mass of Jupiter, that of the Sun being repre sented by I. It is necessary to add unity to the denomination of this fraction, be cause the force which retains Jupiter in his orbit is the sum of the attractions of Jupiter and the Sun. The mass of Jupi ter is • The mass of Saturn and Herschel may be calculated in the same manner. That of the Earth is best determined by the following method : If we take the mean distance of the Earth from the Sun for unity, the arch describ ed by the Earth in a second of time will be the ratio of the circumference to the radius divided by the number of seconds in a sidereal year. If we divide the

square of that arch by the diameter, we obtain its versed sine, which is the deflection of the Earth towards the Sun in a second. But on that parallel of the Earth's surface, the square of the sine of whose latitude is a body falls in a second feet. To reduce this at traction to the mean distance of the Earth from the Sun, we must divide the num ber by the feet contained in that distance ; but the radius of the Earth at the above. mentioned parallel is 19,614,648 French feet. If we divide this number by the tangent of the solar parallax, we obtain the mean radius of the Earth's orbit ex pressed in feet. The effect of the at. traction of the Earth, at a distance equal to the mean radius of its orbit, is equal to 161 L....multiplied by the cube of the IT6T4648 1479560.5.

tangent of the solar parallax = E.P.

Hence the masses of the Sun and Earth are to each other as the numbers 14.79560.5 and 4.486113 ; therefore the 1 mass of the Earth is 3298uy that of the Sun being unity. M. de la Place calcu lated the masses of Mars and Venus from the secular diminution of the obliquity of the ecliptic, and from the mean accelera tion of the Moon's motion. The mass of Mercury he obtained from its volume, supposing the densities of that planet and of the Earth reciprocally as their mean distance from the Sun, a rule which holds with respect to the Earth, Jupiter, and Saturn. The following table exhibits the masses of the different planets, that of the Sun being unity : Mercury 1 Z•Vo4JOIV Venus 1 383137 Earth 1 329809 Mars 1 1846082 Jupiter 1 1067.09 Saturn 1 3359.40 Iferschel 1 19304 The densities of bodies are proportion al to their masses divided by their bulks ; and when . bodies are nearly spherical, their bulks are as the cubes of their semi diameters, of course the densities in that case are as the masses divided by the cubes of the semi-diameters.