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the third day it will move from b to c, making the area b S c equal to A Sa or a S b, and so on.

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(31.) Upon this principle mathematicians have invented methods of calculating the place of a planet, or satellite, at any time for which it may be required. These methods are too troublesome for us to explain here; but we may point out the meaning of two terms which are frequently used in these computations. Suppose, for instance, as in the figure, that the planet, or satellite, occupies ten days in describing the half of its orbit, A a b c d e f g h i B, or twenty days in describing the whole orbit; and suppose that we wished to find its place at the end of three days after leaving the perihelion. If the orbit were a circle, the planet would in three days have moved through an angle of 54 degrees. If the excentricity of the orbit were small (that is, if the orbit did not differ much from a circle), the angle through which the planet would have moved would not differ much from 54 degrees. The excentricities of all the orbits of the planets are small; and it is convenient, therefore, to begin with the angle of 54 degrees as one which is not very erroneous, but which will require some correction. This angle (as 54 degrees), which is propor

tional to the time, is called the mean anomaly; and the correction which it requires, in order to produce the true anomaly, is called the equation of the centre. If we examine the nature of the motion, while the planet moves from A to B, it will readily be seen, that, during the whole of that time, the angle really described by the planet is greater than the angle which is proportional to the time, or the equation of the centre is to be added to the mean anomaly, in order to produce the true anomaly; but while the planet moves in the other half of the orbit, from B to A, the angle really described by the planet is less than the angle which is proportional to the time, or the equation of the centre is to be subtracted from the mean anomaly, in order to produce the true anomaly.

(32.) The sum of the mean anomaly and the longitude of perihelion is called the mean longitude of the planet. It is evident, that if we add the equation of the centre to the mean longitude, while the planet is moving from A to B, or subtract it from the mean longitude, while the planet is moving from B to A, as in (31.), we shall form the true longitude.

(33.) The reader will see, that when the planet's true anomaly is calculated, the length of the radius vector can be computed from a knowledge of the properties of the ellipse. Thus the place of the planet, for any time, is perfectly known. This problem has acquired considerable celebrity under the name of Kepler's problem.

(34.) There remains only one point to be explained

regarding the undisturbed motion of planets and satellites; namely, the relation between a planet's periodic time and the dimensions of the orbit in which it

moves.

Now, on the law of gravitation it has been demonstrated from theory, and it is fully confirmed by observation, that the periodic time does not depend on the excentricity, or on the perihelion distance, or on the aphelion distance, or on any element except the mean distance or semi-major axis. So that if two planets moved round the sun, one in a circle, or in an orbit nearly circular, and the other in a very flat ellipse; provided their mean distances were equal, their periodic times would be equal. It is demonstrated also, that for planets at different distances, the relation between the periodic times and the mean distances is the following:-The squares of the numbers of days (or hours, or minutes, &c.) in the periodic times have the same proportion as the cubes of the numbers of miles (or feet, &c.) in the mean distances.

(35.) Thus the periodic time of Jupiter round the sun is 4332-7 days, and that of Saturn is 10759-2 days; the squares of these numbers are 18772289 and 115760385. The mean distance of Jupiter from the sun is about 487491000 miles, and that of Saturn is about 893955000 miles; the cubes of these numbers are 1158496 (20 ciphers), and 7144088 (20 ciphers). On trial it will be found, that 18772289 and 115760385 are in almost exactly the same proportion as 1158496 and 7144088.

(36.) In like manner, the periodic times of Jupiter's third and fourth satellites round Jupiter are 7.15455 and 16-68877 days; the squares of these numbers are 51.1876 and 278.515. Their mean distances from Jupiter are 670080 and 1178560 miles; the cubes of these numbers are 300866 (12 ciphers), and 1637029 (12 ciphers), and the proportion of 51.1876 to 278-515 is almost exactly the same as the proportion of 300866 to 1637029.

(37.) It must, however, be observed that this rule applies in comparing the periodic times and mean distances, only of bodies which revolve round the same central body. Thus the rule applies in comparing the periodic times and mean distances of Jupiter and Saturn, because they both revolve round the sun; it applies in comparing the periodic times and mean distances of Jupiter's third and fourth satellites, because they both revolve round Jupiter; but it would not apply in comparing the periodic time and mean distance of Saturn revolving round the sun with that of Jupiter's third satellite revolving round Jupiter.

(38.) In comparing the orbits described by different planets, or satellites, round different centres of force, theory gives us the following law:-The cubes of the mean distances are in the same proportion as the products of the mass by the square of the periodic time. Thus, for instance, the mean distance of Jupiter's fourth satellite from Jupiter is 1178560 miles; its periodic time round Jupiter is 16-68877 days; the mean distance of the earth from the sun is 93726900 miles; its

periodic time round the sun is 365 2564 days; also the mass of Jupiter is Toth the sun's mass. The cubes of the mean distances are respectively 1637029 (12 ciphers), and 823365 (18 ciphers); the products of the squares of the times by the masses are respectively 0.265252 and 133412; and these numbers are in the same proportion as 1637029 (12 ciphers), and 823365 (18 ciphers).

(39.) The three rules, that planets move in ellipses; that the radius vector in each orbit passes over areas proportional to the times, and that the squares of the periodic times are proportional to the cubes of the mean distances, are commonly called Kepler's laws. They were discovered by Kepler from observation, before the theory of gravitation was invented; they were first explained from the theory by Newton, about A.D. 1680.

(40.) The last of these is not strictly true, unless we suppose that the central body is absolutely immovable. This, however, is evidently inconsistent with the principles which we have laid down in Section I. In considering the motion, for instance, of Jupiter round the sun, it is necessary to consider, that, while the sun attracts Jupiter, Jupiter is also attracting the sun. But the planets are so small in comparison with the sun (the largest of them, Jupiter, having less than onethousandth part of the matter contained in the sun), that in common illustrations there is no need to take this consideration into account. For nice astronomical purposes it is taken into account in the following

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