212. Remarkable relation between the inclinations of several 214. Explanation of the term fundamental plane 220. Their lines of nodes will regress on their fundamental planes, without alteration or inclination to those planes. 155 221. An inclination of the orbit of an exterior satellite produces a similar inclination of the orbit of an interior satellite . 156 222. An inclination of the orbit of an interior satellite produces 230. The attraction of an oblate spheroid on a body in the plane 233. The attraction of an oblate spheroid on a body in the direc- 236. The attraction of an oblate spheroid causes the lines of apses of its satellites to progress; and more rapidly for the near satellites than for the distant ones. 238. If a satellite is not in the plane of the equator, the attraction 240. Effect of Saturn's ring on the motions of Saturn's satellites. 170 242. The moon's fundamental plane is inclined to the ecliptic ON GRAVITATION. SECTION I.-On the Rules for calculating Attraction, or the Law of Gravitation. (1.) THE principle upon which the motions of the earth, moon, and planets are calculated is this: Every particle of matter attracts every other particle. That is, if there were a single body alone, and at rest, then, if a second body were brought near it, the first body would immediately begin to move towards the second body. Just in the same manner, if a needle is at rest on a table, and if a magnet is brought near it, the needle immediately begins to move towards the magnet, and we say that the magnet attracts the needle. But magnetic attraction belongs only to certain bodies; whereas the attraction of which we speak here belongs to all bodies of every kind: metals, earths, fluids, and even the air and gases are equally subject to its influence. (2.) The most remarkable experiments which prove that bodies attract each other are a set of experiments made at the end of the last century by Mr. Cavendish. Small leaden balls were supported on the ends of a rod which was suspended at the middle by a slender B wire; and when large leaden balls were brought near them, it was found that the wire was immediately twisted by the motion of the balls. But the results of this experiment are striking, principally because they are unusual; the ordinary force of gravity serves quite as well to prove the existence of some such power. For when we consider that the earth is round, and that, on all parts of it, bodies, as soon as they are at liberty, fall in directions perpendicular to its surface (and therefore fall in opposite directions at the places which are diametrically opposite), we are compelled to allow that there is a force such as we call attraction, either directed to the centre of the earth, or produced by a great number of small forces, directed to all the different particles composing the earth. The peculiar value of Cavendish's experiment consists in showing that there is a small force directed to every different particle of the earth. (3.) But it is necessary to state distinctly the rules by which this attraction is regulated, and by which it may be calculated; or (as it is technically called) the law of gravitation. Before we can do this, we must determine which of the effects of attraction we choose to take as its measure. For there are two distinct effects one is the pressure which it produces upon any obstacle that keeps the body at rest; the other is the space through which it draws the body in a certain time, if the obstacle is removed and the body set at liberty, Thus, to take the ordinary force of gravity as an instance: we might measure it by the pressure which is produced on the hand by a lump of lead held in the hand; or we might measure it by the number of inches through which the lump of lead would fall in a second of time after the hand is opened (as the pressure and the fall are both occasioned by gravity). But there is this difference between the two measures; if we adopted the first, since a large lump of lead weighs more than a small one, we should find a different measure by the use of every different piece of lead; whereas, if we adopt the second, since it is well established by careful and accurate experiments that large and small lumps of lead, stones, and even feathers, fall through the same number of inches in second of time (when the resistance of the air, &c., is removed), we shall get the same measure for gravity, whatever body we suppose subject to its influence. The consistence and simplicity of the measure thus obtained incline us to adopt it in every other case; and thus we shall say, Attraction is measured by the space through which it draws a body in one second of time after the body is set at liberty. (4.) Whenever we speak, therefore, of calculating attraction, it must be understood to mean calculating the number of inches, or feet, through which the attraction draws the body in one second of time. (5.) Now the first rule is this: "The attraction of one body upon another body does not depend on the mass of the body which is attracted, but is the same whatever be the mass of the body so attracted, if the distances are the same." (6.) Thus Jupiter attracts the sun, and Jupiter attracts the earth also; but though the sun's mass is three hundred thousand times as great as the earth's, yet the attraction of Jupiter on the sun is exactly equal to his attraction on the earth, when the sun and the earth are equally distant from Jupiter. In other words (the attraction being measured in conformity with the definition above), when the sun and the earth are at equal distances from Jupiter, the attraction of Jupiter draws the sun through as many inches, or parts of an inch, in one second of time as it draws the earth in the same time. (7.) The second rule is this: "Attraction is proportional to the mass of the body which attracts, if the distances of different attracting bodies be the same.” (8.) Thus, suppose that the sun and Jupiter are at equal distances from Saturn; the sun is about a thousand times as big as Jupiter; then whatever be the number of inches through which Jupiter draws Saturn in one second of time, the sun draws Saturn in the same time through a thousand times that number of inches. (9.) The third rule is this: "If the same attracting body act upon several bodies at different distances, the attractions are inversely proportional to the square of the distances from the attracting body." (10.) Thus the earth attracts the sun, and the earth also attracts the moon; but the sun is four hundred times as far off as the moon, and therefore, the earth's attraction on the sun is only 16th part of its attraction on the moon; or, as the earth's attraction draws the moon through about 1th of an inch in one second of time, the earth's attraction draws the sun |